Download - 11X1 T09 01 limits and continuity (2010)
Limits & Continuity
Limits & ContinuityA limit describes the behaviour of functions.
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
0
lim 4x
f x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
0
lim 4x
f x
lim :x a
f x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
0
lim 4x
f x
lim :x a
f x
as the x value approaches a from the positive side, what value does f(x) approach?
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
0
lim 4x
f x
lim :x a
f x
as the x value approaches a from the positive side, what value does f(x) approach?
1lim 1 0x
x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
0
lim 4x
f x
lim :x a
f x
as the x value approaches a from the positive side, what value does f(x) approach?
1lim 1 0x
x
0
lim 6x
f x
Limits & ContinuityA limit describes the behaviour of functions.
lim :x a
f x
as the x value approaches a, what value does f(x) approach?
lim :x a
f x
as the x value approaches a from the negative side, what value does f(x) approach?
y
x1
1
1y x
1lim 1 0x
x
y
x
46
y f x
0
lim 4x
f x
lim :x a
f x
as the x value approaches a from the positive side, what value does f(x) approach?
1lim 1 0x
x
0
lim 6x
f x
If lim lim , then is continuous at x a x a
f x f x f x x a
Finding Limits
Finding Limits(1) Direct Substitution
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
5 7
12
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
5 7
12
(2) Factorise and Cancel
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
5 7
12
(2) Factorise and Cancel2
3
9e.g. lim3x
xx
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
5 7
12
(2) Factorise and Cancel2
3
9e.g. lim3x
xx
3
3 3lim
3x
x xx
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
5 7
12
(2) Factorise and Cancel2
3
9e.g. lim3x
xx
3
3 3lim
3x
x xx
3
lim 3x
x
Finding Limits(1) Direct Substitution
5e.g. lim 7
xx
5 7
12
(2) Factorise and Cancel2
3
9e.g. lim3x
xx
3
3 3lim
3x
x xx
3
lim 3x
x
3 36
(3) Special Limit
(3) Special Limit1lim 0
x x
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
2 3
3
3 2 11lim 14x
x x x
x
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
2 3
3
3 2 11lim 14x
x x x
x
14
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
2 3
3
3 2 11lim 14x
x x x
x
14
2
34 lim
1x
x xiix
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
2 3
3
3 2 11lim 14x
x x x
x
14
2
34 lim
1x
x xiix
010
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
2 3
3
3 2 11lim 14x
x x x
x
14
2
34 lim
1x
x xiix
010
7 6 2
7 lim3 974x
x x xiiix x
(3) Special Limit1lim 0
x x
3 2
33 2 1e.g. lim4 1x
x x xix
3 2
3 3 3 3
3
3 3
3 2 1
lim4 1x
x x xx x x x
xx x
2 3
3
3 2 11lim 14x
x x x
x
14
2
34 lim
1x
x xiix
010
7 6 2
7 lim3 974x
x x xiiix x
13
3
22 lim1x
xivx
3
22 lim1x
xivx
10
3
22 lim1x
xivx
10
3 2 Find the horizontal asymptote of
1 1x x
v yx x
3
22 lim1x
xivx
10
3 2 Find the horizontal asymptote of
1 1x x
v yx x
3 2lim
1 1x
x xx x
3
22 lim1x
xivx
10
3 2 Find the horizontal asymptote of
1 1x x
v yx x
3 2lim
1 1x
x xx x
2
26lim
1x
x xx
3
22 lim1x
xivx
10
3 2 Find the horizontal asymptote of
1 1x x
v yx x
3 2lim
1 1x
x xx x
2
26lim
1x
x xx
111
3
22 lim1x
xivx
10
3 2 Find the horizontal asymptote of
1 1x x
v yx x
3 2lim
1 1x
x xx x
2
26lim
1x
x xx
111
horizontal asymptote is 1y
3
22 lim1x
xivx
10
3 2 Find the horizontal asymptote of
1 1x x
v yx x
3 2lim
1 1x
x xx x
2
26lim
1x
x xx
111
horizontal asymptote is 1y Exercise 7I; 1a, 2ace, 3ac,
4a, 5ad, 8a, 9ab, 10a
