lesson 6: limits involving infinity (section 41 handout)
TRANSCRIPT
Section 1.6Limits involving Infinity
V63.0121.041, Calculus I
New York University
September 21, 2010
Announcements
I
Announcements
I
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 2 / 35
Objectives
I “Intuit” limits involvinginfinity by eyeballing theexpression.
I Show limits involving infinityby algebraic manipulationand conceptual argument.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 3 / 35
Notes
Notes
Notes
1
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Recall the definition of limit
Definition
We writelimx→a
f (x) = L
and say
“the limit of f (x), as x approaches a, equals L”
if we can make the values of f (x) arbitrarily close to L (as close to L as welike) by taking x to be sufficiently close to a (on either side of a) but notequal to a.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 4 / 35
Recall the unboundedness problem
Recall why limx→0+
1
xdoesn’t exist.
x
y
L?
No matter how thin we draw the strip to the right of x = 0, we cannot“capture” the graph inside the box.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 5 / 35
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limit forms
Limits at ∞Algebraic rates of growthRationalizing to get a limit
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 6 / 35
Notes
Notes
Notes
2
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Infinite Limits
Definition
The notation
limx→a
f (x) =∞
means that values of f (x) can bemade arbitrarily large (as large aswe please) by taking x sufficientlyclose to a but not equal to a.
I “Large” takes the place of“close to L”.
x
y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 7 / 35
Negative Infinity
Definition
The notationlimx→a
f (x) = −∞
means that the values of f (x) can be made arbitrarily large negative (aslarge as we please) by taking x sufficiently close to a but not equal to a.
I We call a number large or small based on its absolute value. So−1, 000, 000 is a large (negative) number.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 8 / 35
Vertical Asymptotes
Definition
The line x = a is called a vertical asymptote of the curve y = f (x) if atleast one of the following is true:
I limx→a
f (x) =∞I lim
x→a+f (x) =∞
I limx→a−
f (x) =∞
I limx→a
f (x) = −∞I lim
x→a+f (x) = −∞
I limx→a−
f (x) = −∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 9 / 35
Notes
Notes
Notes
3
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Infinite Limits we Know
I limx→0+
1
x=∞
I limx→0−
1
x= −∞
I limx→0
1
x2=∞
x
y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 10 / 35
Finding limits at trouble spots
Example
Let
f (x) =x2 + 2
x2 − 3x + 2
Find limx→a−
f (x) and limx→a+
f (x) for each a at which f is not continuous.
Solution
The denominator factors as (x − 1)(x − 2). We can record the signs of thefactors on the number line.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 11 / 35
Use the number line
(x − 1)−
1
0 +
(x − 2)−
2
0 +
(x2 + 2)+
f (x)1 2
+ +∞ −∞ − −∞ +∞ +
Solim
x→1−f (x) = +∞ lim
x→2−f (x) = −∞
limx→1+
f (x) = −∞ limx→2+
f (x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 12 / 35
Notes
Notes
Notes
4
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
In English, now
To explain the limit, you can say:“As x → 1−, the numerator approaches 3, and the denominatorapproaches 0 while remaining positive. So the limit is +∞.”
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 13 / 35
The graph so far
limx→1−
f (x) = +∞ limx→2−
f (x) = −∞
limx→1+
f (x) = −∞ limx→2+
f (x) = +∞
x
y
−1 1 2 3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 14 / 35
Rules of Thumb with infinite limits
Fact
I If limx→a
f (x) =∞ and
limx→a
g(x) =∞, then
limx→a
(f (x) + g(x)) =∞.
∞+∞ =∞
I If limx→a
f (x) = −∞ and
limx→a
g(x) = −∞, then
limx→a
(f (x) + g(x)) = −∞.
−∞−∞ =∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 15 / 35
Notes
Notes
Notes
5
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Rules of Thumb with infinite limits
Fact
I If limx→a
f (x) = L and limx→a
g(x) = ±∞,
L +∞ =∞L−∞ = −∞
then
limx→a
(f (x) + g(x)) = ±∞.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 16 / 35
Rules of Thumb with infinite limitsKids, don’t try this at home!
Fact
I The product of a finite limit and an infinite limit is infinite if the finitelimit is not 0.
L · ∞ =
{∞ if L > 0
−∞ if L < 0.
L · (−∞) =
{−∞ if L > 0
∞ if L < 0.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 17 / 35
Multiplying infinite limitsKids, don’t try this at home!
Fact
I The product of two infinite limits is infinite.
∞ ·∞ =∞∞ · (−∞) = −∞
(−∞) · (−∞) =∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 18 / 35
Notes
Notes
Notes
6
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Dividing by InfinityKids, don’t try this at home!
Fact
I The quotient of a finite limit by an infinite limit is zero.
L
∞= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 19 / 35
Dividing by zero is still not allowed
1
0=∞
There are examples of such limit forms where the limit is ∞, −∞,undecided between the two, or truly neither.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 20 / 35
Indeterminate Limit forms
Limits of the formL
0are indeterminate. There is no rule for evaluating
such a form; the limit must be examined more closely. Consider these:
limx→0
1
x2=∞ lim
x→0
−1
x2= −∞
limx→0+
1
x=∞ lim
x→0−
1
x= −∞
Worst, limx→0
1
x sin(1/x)is of the form
L
0, but the limit does not exist, even
in the left- or right-hand sense. There are infinitely many verticalasymptotes arbitrarily close to 0!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 21 / 35
Notes
Notes
Notes
7
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Indeterminate Limit forms
Limits of the form 0 · ∞ and ∞−∞ are also indeterminate.
Example
I The limit limx→0+
sin x · 1
xis of the form 0 · ∞, but the answer is 1.
I The limit limx→0+
sin2 x · 1
xis of the form 0 · ∞, but the answer is 0.
I The limit limx→0+
sin x · 1
x2is of the form 0 · ∞, but the answer is ∞.
Limits of indeterminate forms may or may not “exist.” It will depend onthe context.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 22 / 35
Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 23 / 35
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limit forms
Limits at ∞Algebraic rates of growthRationalizing to get a limit
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 24 / 35
Notes
Notes
Notes
8
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Definition
Let f be a function defined on some interval (a,∞). Then
limx→∞
f (x) = L
means that the values of f (x) can be made as close to L as we like, bytaking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve y = f (x)if either
limx→∞
f (x) = L or limx→−∞
f (x) = L.
y = L is a horizontal line!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 25 / 35
Basic limits at infinity
Theorem
Let n be a positive integer. Then
I limx→∞
1
xn= 0
I limx→−∞
1
xn= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 26 / 35
Using the limit laws to compute limits at ∞
Example
Find limx→∞
x
x2 + 1
Answer
The limit is 0.
x
y
Notice that the graph does cross the asymptote, which contradicts one ofthe heuristic definitions of asymptote.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 27 / 35
Notes
Notes
Notes
9
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Solution
Solution
Factor out the largest power of x from the numerator and denominator.We have
x
x2 + 1=
x(1)
x2(1 + 1/x2)=
1
x· 1
1 + 1/x2
limx→∞
x
x2 + 1= lim
x→∞
1
x
1
1 + 1/x2= lim
x→∞
1
x· limx→∞
1
1 + 1/x2
= 0 · 1
1 + 0= 0.
Remark
Had the higher power been in the numerator, the limit would have been∞.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 28 / 35
Another Example
Example
Find
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 29 / 35
Solution
Solution
Factor out the largest power of x from the numerator and denominator.We have
2x3 + 3x + 1
4x3 + 5x2 + 7=
x3(2 + 3/x2 + 1/x3)
x3(4 + 5/x + 7/x3)
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7= lim
x→∞
2 + 3/x2 + 1/x3
4 + 5/x + 7/x3
=2 + 0 + 0
4 + 0 + 0=
1
2
Upshot
When finding limits of algebraic expressions at infinity, look at the highestdegree terms.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 30 / 35
Notes
Notes
Notes
10
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Still Another Example
Example
Find
limx→∞
√3x4 + 7
x2 + 3
√3x4 + 7 ∼
√3x4 =
√3x2
Answer
The limit is√
3.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 31 / 35
Solution
Solution
limx→∞
√3x4 + 7
x2 + 3= lim
x→∞
√x4(3 + 7/x4)
x2(1 + 3/x2)
= limx→∞
x2√
(3 + 7/x4)
x2(1 + 3/x2)
= limx→∞
√(3 + 7/x4)
1 + 3/x2
=
√3 + 0
1 + 0=√
3.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 32 / 35
Rationalizing to get a limit
Example
Compute limx→∞
(√4x2 + 17− 2x
).
Solution
This limit is of the form ∞−∞, which we cannot use. So we rationalizethe numerator (the denominator is 1) to get an expression that we can usethe limit laws on.
limx→∞
(√4x2 + 17− 2x
)= lim
x→∞
(√4x2 + 17− 2x
)·√
4x2 + 17 + 2x√4x2 + 17 + 2x
= limx→∞
(4x2 + 17)− 4x2
√4x2 + 17 + 2x
= limx→∞
17√4x2 + 17 + 2x
= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 33 / 35
Notes
Notes
Notes
11
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010
Kick it up a notch
Example
Compute limx→∞
(√4x2 + 17x − 2x
).
Solution
Same trick, different answer:
limx→∞
(√4x2 + 17x − 2x
)= lim
x→∞
(√4x2 + 17x − 2x
)·√
4x2 + 17 + 2x√4x2 + 17x + 2x
= limx→∞
(4x2 + 17x)− 4x2
√4x2 + 17x + 2x
= limx→∞
17x√4x2 + 17x + 2x
= limx→∞
17√4 + 17/x + 2
=17
4
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 34 / 35
Summary
I Infinity is a more complicated concept than a single number. Thereare rules of thumb, but there are also exceptions.
I Take a two-pronged approach to limits involving infinity:I Look at the expression to guess the limit.I Use limit rules and algebra to verify it.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 21, 2010 35 / 35
Notes
Notes
Notes
12
Section 1.6 : Limits involving InfinityV63.0121.041, Calculus I September 21, 2010