limit of a function and one-sided limits
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Limit of a Function and
One-sided limits
Mathematics 53
Institute of Mathematics (UP Diliman)
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
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Introduction
Given a function f(x)anda
,
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Introduction
Given a function f(x)anda
,
what is the value of f atxneara,
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Introduction
Given a function f(x)anda
,
what is the value of f atxneara,
but not equal toa?
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Illustration 1
Consider f(x) =3x 1.
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 1
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 5
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
Based on the table, as x gets closer and closer to 1,
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
Based on the table, as x gets closer and closer to 1, f(x)gets closer and closer
to2.
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Ill i 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill t ti 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill t ti 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill stration 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
Asxgets closer and closer to 1, f(x)gets closer and closer to 2.
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Illustration 2
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Illustration 2
Consider: g(x) = 3x2 4x+1
x 1
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Illustration 2
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Illustration 2
Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
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Illustration 2
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Illustration 2
Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1, h(x)gets closer and closer to 2.
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Limit
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Limit
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Intuitive Notion of a Limit
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Limit
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Intuitive Notion of a Limita ,L
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
The limit of f(x)asxapproachesa isL
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
The limit of f(x)
asx
approachesa
isL
if the values of f(x)get closer and closer to Lasxassumes values getting closer
and closer toabut not reachinga.
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
The limit of f(x)
asx
approachesa
isL
if the values of f(x)get closer and closer to Lasxassumes values getting closer
and closer toabut not reachinga.
Notation:limxa f(x) = L
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1)
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1) =2
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1) =2
Note: In this case, limx1
f(x)
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1) =2
Note: In this case, limx1
f(x) = f(1).
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g(x) = 3
x
2
4
x+1
x 1
1 1 2 31
1
2
3
4
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g(x) = 3
x
2
4
x+1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x+1x 1
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g(x) = 3x2
4x+1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x+1x 1 =2
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g(x) = 3x2
4x+1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x+1x 1 =2
Note: Thoughg(1)is undefined,limx1
g(x)exists.
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
limx1
h(x)
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
limx1
h(x) =2
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
limx1
h(x) =2
Note:h(1) = limx1 h(x).
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close toabut not exactly at a.
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Remark
In finding limxa f(x):
We only need to consider values of x very close toabut not exactly at a.
Thus, limxa f(x)is NOT NECESSARILYthe same as f(a).
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Remark
In finding limxa f(x):
We only need to consider values of x very close toabut not exactly at a.
Thus, limxa f(x)is NOT NECESSARILYthe same as f(a).
We letxapproachafrom BOTH SIDES ofa.
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If f(x)
does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
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If f(x)
does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
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If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
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Some Remarks
-
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77/289
If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0?
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
-
8/12/2019 Limit of a Function and One-sided Limits
78/289
If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
-
8/12/2019 Limit of a Function and One-sided Limits
79/289
If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1?
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
-
8/12/2019 Limit of a Function and One-sided Limits
80/289
If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
-
8/12/2019 Limit of a Function and One-sided Limits
81/289
If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
limx0
H(x) dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
For today
-
8/12/2019 Limit of a Function and One-sided Limits
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1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 16 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
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Theorem
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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84/289
Theorem
If limxa f(x)exists, then it is unique.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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85/289
Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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8/12/2019 Limit of a Function and One-sided Limits
86/289
Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c=c.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
87/289
Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c=c.limxa x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
88/289
Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c=c.limxa x=a
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
89/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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90/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
91/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
92/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
93/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
94/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
95/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
-
8/12/2019 Limit of a Function and One-sided Limits
96/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Th
-
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97/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2limxa[c f(x)] =
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Th
-
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98/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2limxa[c f(x)] =climxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Th
-
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99/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2limxa[c f(x)] =climxa f(x) =cL1
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
-
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100/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
-
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101/289
Limit Theorems
Theorem
-
8/12/2019 Limit of a Function and One-sided Limits
102/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
-
8/12/2019 Limit of a Function and One-sided Limits
103/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
-
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104/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
limxa (f(x))
n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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105/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
limxa (f(x))
n =
limxa f(x)
n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
-
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106/289
Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
limxa (f(x))
n =
limxa f(x)
n = (L1)n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
-
8/12/2019 Limit of a Function and One-sided Limits
107/289
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4)
-
8/12/2019 Limit of a Function and One-sided Limits
108/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2
-
8/12/2019 Limit of a Function and One-sided Limits
109/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
-
8/12/2019 Limit of a Function and One-sided Limits
110/289
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x
-
8/12/2019 Limit of a Function and One-sided Limits
111/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x
-
8/12/2019 Limit of a Function and One-sided Limits
112/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
-
8/12/2019 Limit of a Function and One-sided Limits
113/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
-
8/12/2019 Limit of a Function and One-sided Limits
114/289
=2
limx1 x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
-
8/12/2019 Limit of a Function and One-sided Limits
115/289
=2
limx1 x2
+3
limx1 x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
-
8/12/2019 Limit of a Function and One-sided Limits
116/289
=2
limx1 x2
+3
limx1 x limx1 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
-
8/12/2019 Limit of a Function and One-sided Limits
117/289
=2
limx1 x2
+3
limx1 x limx1 4
=2
limx1
x
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
-
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118/289
=2
limx1 x2
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
-
8/12/2019 Limit of a Function and One-sided Limits
119/289
=2
limx1 x2
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
-
8/12/2019 Limit of a Function and One-sided Limits
120/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
-
8/12/2019 Limit of a Function and One-sided Limits
121/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
-
8/12/2019 Limit of a Function and One-sided Limits
122/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
l2
l l
-
8/12/2019 Limit of a Function and One-sided Limits
123/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
-
8/12/2019 Limit of a Function and One-sided Limits
124/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
= 5
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
-
8/12/2019 Limit of a Function and One-sided Limits
125/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
= 5
In general:
Remark
If f is a polynomial function, then limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
-
8/12/2019 Limit of a Function and One-sided Limits
126/289
=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
= 5
In general:
Remark
If f is a polynomial function, then limxa f(x) = f(a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
-
8/12/2019 Limit of a Function and One-sided Limits
127/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
li 4x3 + 3x2 x + 1
-
8/12/2019 Limit of a Function and One-sided Limits
128/289
limx2
4x3 +3x2 x+1x2 +2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
li 4x3
+ 3x2
x + 1lim
x 2(4x3 +3x2 x+1)
-
8/12/2019 Limit of a Function and One-sided Limits
129/289
limx2
4x +3x x+1x2 +2
= x2 lim
x2(x2 +2)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
li 4x3
+ 3x2
x + 1lim
x2(4x3 +3x2 x+1)
-
8/12/2019 Limit of a Function and One-sided Limits
130/289
limx2
4x +3x x+1x2 +2
= x2 lim
x2(x2 +2)
= 4(8)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1lim
x2(4x3 +3x2 x+1)
-
8/12/2019 Limit of a Function and One-sided Limits
131/289
limx2
4x +3x x+1x2 +2
= x 2 lim
x2(x2 +2)
= 4(8) +3(4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
-
8/12/2019 Limit of a Function and One-sided Limits
132/289
limx2
4x +3x x+1x2 +2
= x 2 lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
-
8/12/2019 Limit of a Function and One-sided Limits
133/289
limx2
4x +3x x+1x2 +2
= x 2lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
-
8/12/2019 Limit of a Function and One-sided Limits
134/289
limx2
4x +3x x+1x2 +2
=lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
= 176
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
-
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135/289
limx2
4x +3x x+1x2 +2
=lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
= 176
Remark
If fis a rational function and f(a)is defined,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 12
=lim
x2(4x3 +3x2
x+1)
2
-
8/12/2019 Limit of a Function and One-sided Limits
136/289
limx2
4x +3x x+1x2 +2
lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
= 176
Remark
If fis a rational function and f(a)is defined, then lim
xaf(x) = f(a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxan
f(x) = n
limxa f(x),
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137/289
f ( )
f ( ),
provided limxa f(x) > 0whenn is even.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxan
f(x) = n
limxa f(x),
-
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138/289
f ( )
f ( ),
provided limxa f(x) > 0whenn is even.
limx3
3x
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
-
8/12/2019 Limit of a Function and One-sided Limits
139/289
f ( )
f ( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
-
8/12/2019 Limit of a Function and One-sided Limits
140/289
f ( )
f ( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1) =
8
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
-
8/12/2019 Limit of a Function and One-sided Limits
141/289
f ( )
f ( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1) =
8= 2
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
-
8/12/2019 Limit of a Function and One-sided Limits
142/289
( )
( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1) =
8= 2
2
limx1
3
x+4
x 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
-
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143/289
provided lim
xa f(x) > 0whenn is even.
limx3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
-
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144/289
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf
(x
) = nlimxa f
(x
),
-
8/12/2019 Limit of a Function and One-sided Limits
145/289
provided lim
xa f(x) > 0whenn is even.
limx3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
-
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146/289
provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
-
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147/289
provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
-
8/12/2019 Limit of a Function and One-sided Limits
148/289
provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
-
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149/289
provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4=??
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
-
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150/289
provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4=?? (for now)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
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151/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
-
8/12/2019 Limit of a Function and One-sided Limits
152/289
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
-
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153/289
=
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
-
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154/289
=
limx3
2x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
-
8/12/2019 Limit of a Function and One-sided Limits
155/289
=
limx3
2x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
-
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156/289
=
limx3
2x2
limx3
(5x+1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
-
8/12/2019 Limit of a Function and One-sided Limits
157/289
=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
-
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=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
-
8/12/2019 Limit of a Function and One-sided Limits
159/289
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
-
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160/289
=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18 4
28
3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
-
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161/289
=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18 4
28
3
= 1
8
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
-
8/12/2019 Limit of a Function and One-sided Limits
162/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
-
8/12/2019 Limit of a Function and One-sided Limits
163/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
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164/289
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0
-
8/12/2019 Limit of a Function and One-sided Limits
165/289
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1)
-
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166/289
x1
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
-
8/12/2019 Limit of a Function and One-sided Limits
167/289
x1
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
-
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x x
But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
-
8/12/2019 Limit of a Function and One-sided Limits
169/289
But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
-
8/12/2019 Limit of a Function and One-sided Limits
170/289
But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
limx1
3x2 4x+1x 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
-
8/12/2019 Limit of a Function and One-sided Limits
171/289
But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
limx1
3x2 4x+1x 1 = limx1(3x 1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
-
8/12/2019 Limit of a Function and One-sided Limits
172/289
But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
limx1
3x2 4x+1x 1 = limx1(3x 1) =2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Definition
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0
-
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174/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
lim
xa
f(x)
g(x)
is called anindeterminate form of type 0
0.
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175/289
yp0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
lim
xa
f(x)
g(x)
is called anindeterminate form of type 0
0.
-
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176/289
yp0
Remarks:
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
-
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
-
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
3 Some techniques used in evaluating such limits are:
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
-
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
3 Some techniques used in evaluating such limits are:
FactoringRationalization
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
-
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181/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0
-
8/12/2019 Limit of a Function and One-sided Limits
182/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0
limx 1
x2 +2x+1
x + 1
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183/289
x1 x+1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x 1x2 +2x+1
x + 1 = lim
x 1
-
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184/289
x1 x+1 x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x + 1 = lim
x1(x+1)2
-
8/12/2019 Limit of a Function and One-sided Limits
185/289
x 1 x+1 x 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x + 1 = lim
x1(x+1)2
x + 1
-
8/12/2019 Limit of a Function and One-sided Limits
186/289
x 1 x+1 x 1 x+1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x+1 = lim
x1(x+1)2
x+1
-
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187/289
x 1 x + 1 x 1 x + 1
= limx
1(x+1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: limx
1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x+1 = lim
x1(x+1)2
x+1
-
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+ +
= limx
1(x+1)
= (1+1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: limx
1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x+1 = lim
x1(x+1)2
x+1
-
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= limx
1(x+1)
= (1+1)
= 0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
-
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190/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0
-
8/12/2019 Limit of a Function and One-sided Limits
191/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4
-
8/12/2019 Limit of a Function and One-sided Limits
192/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2
-
8/12/2019 Limit of a Function and One-sided Limits
193/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)2 2 4
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= limx
2
x2 2x+4x
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)2 2 4
-
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= limx
2
x2 2x+4x
2
= 4+4+4
2 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)2 2 + 4
-
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= limx
2
x2 2x+4x
2
= 4+4+4
2 2= 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx4
x2 162x
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
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Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
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= limx4
(x 4)(x+4)(2+x)4 x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
-
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= limx4
(x 4)(x+4)(2+x)4 x
= limx4
[(x+4)(2+x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
( )( )(
)
-
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= limx4
(x 4)(x+4)(2+x)4 x
= limx4
[(x+4)(2+x)]
= (8)(4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
( )( )(
)
-
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= limx4
(x 4)(x+4)(2+x)4 x
= limx4
[(x+4)(2+x)]
= (8)(4)
= 32
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+4
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
-
8/12/2019 Limit of a Function and One-sided Limits
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
1
-
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= lim
x81
(x+1)( 3x2 +2 3x+4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
1
-
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= lim
x81
(x+1)( 3x2 +2 3x+4)=
1
9(4+4+4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
1
-
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= lim
x81
(x+1)( 3x2 +2 3x+4)=
1
9(4+4+4)
= 1
108
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
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3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 29 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1.
-
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
2
3
4
0
-
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3
2
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
-
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3
2
1
Asx approaches1 through values less than1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
-
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