This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.
The brain never rests. In the absence of external stimuli, fluctuations in cerebral activity reveal an intrinsic structure that mirrors brain function during cognitive tasks.
Learning and comparing multi-subject models of brain functional connecitivityGael Varoquaux
High-level brain function arises through functional interactions. These can be mapped via co-fluctuations in activity observed in functional imaging.
First, I first how spatial maps characteristic of on-going activity in a population of subjects can be learned using multi-subject decomposition models extending the popular Independent Component Analysis. These methods single out spatial atoms of brain activity: functional networks or brain regions. With a probabilistic model of inter-subject variability, they open the door to building data-driven atlases of on-going activity.
Subsequently, I discuss graphical modeling of the interactions between brain regions. To learn highly-resolved large scale individual
graphical models models, we use sparsity-inducing penalizations introducing a population prior that mitigates the data scarcity at the subject-level. The corresponding graphs capture better the community structure of brain activity than single-subject models or group averages.
Finally, I address the detection of connectivity differences between subjects. Explicit group variability models of the covariance structure can be used to build optimal edge-level test statistics. On stroke patients resting-state data, these models detect patient-specific functional connectivity perturbations.
Vector algebra for Steep Slope Models analysisGeoCommunity
Natalia Kolecka: Vector algebra for Steep Slope Models analysis (poster), 9th International Symposium GIS Ostrava, VŠB – Technical Univerzity of Ostrava, from 23rd to 25th January 2012
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 16: Inverse Trigonometric Functions (Section 041 slides)Mel Anthony Pepito
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The brain never rests. In the absence of external stimuli, fluctuations in cerebral activity reveal an intrinsic structure that mirrors brain function during cognitive tasks.
Learning and comparing multi-subject models of brain functional connecitivityGael Varoquaux
High-level brain function arises through functional interactions. These can be mapped via co-fluctuations in activity observed in functional imaging.
First, I first how spatial maps characteristic of on-going activity in a population of subjects can be learned using multi-subject decomposition models extending the popular Independent Component Analysis. These methods single out spatial atoms of brain activity: functional networks or brain regions. With a probabilistic model of inter-subject variability, they open the door to building data-driven atlases of on-going activity.
Subsequently, I discuss graphical modeling of the interactions between brain regions. To learn highly-resolved large scale individual
graphical models models, we use sparsity-inducing penalizations introducing a population prior that mitigates the data scarcity at the subject-level. The corresponding graphs capture better the community structure of brain activity than single-subject models or group averages.
Finally, I address the detection of connectivity differences between subjects. Explicit group variability models of the covariance structure can be used to build optimal edge-level test statistics. On stroke patients resting-state data, these models detect patient-specific functional connectivity perturbations.
Vector algebra for Steep Slope Models analysisGeoCommunity
Natalia Kolecka: Vector algebra for Steep Slope Models analysis (poster), 9th International Symposium GIS Ostrava, VŠB – Technical Univerzity of Ostrava, from 23rd to 25th January 2012
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 16: Inverse Trigonometric Functions (Section 041 slides)Mel Anthony Pepito
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Lesson 9: The Product and Quotient Rules (Section 21 slides)Mel Anthony Pepito
The derivative of a sum of functions is the sum of the derivatives of those functions, but the derivative of a product or a quotient of functions is not so simple. We'll derive and use the product and quotient rule for these purposes. It will allow us to find the derivatives of other trigonometric functions, and derivatives of power functions with negative whole number exponents.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
1. Section 1.6
Limits involving Infinity
V63.0121.021, Calculus I
New York University
September 23, 2010
Announcements
Quiz 1 is next week in recitation. Covers Sections 1.1–1.4
. . . . . .
2. Announcements
Quiz 1 is next week in
recitation. Covers Sections
1.1–1.4
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 2 / 38
3. Objectives
“Intuit” limits involving
infinity by eyeballing the
expression.
Show limits involving
infinity by algebraic
manipulation and
conceptual argument.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 3 / 38
4. Recall the definition of limit
Definition
We write
lim f(x) = L
x→a
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
we like) by taking x to be sufficiently close to a (on either side of a) but
not equal to a.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 4 / 38
5. Recall the unboundedness problem
1
Recall why lim+ doesn’t exist.
x→0 x
y
.
.?.
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we cannot
“capture” the graph inside the box.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
6. Recall the unboundedness problem
1
Recall why lim+ doesn’t exist.
x→0 x
y
.
.?.
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we cannot
“capture” the graph inside the box.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
7. Recall the unboundedness problem
1
Recall why lim+ doesn’t exist.
x→0 x
y
.
.?.
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we cannot
“capture” the graph inside the box.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
8. Recall the unboundedness problem
1
Recall why lim+ doesn’t exist.
x→0 x
y
.
.?.
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we cannot
“capture” the graph inside the box.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
9. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 6 / 38
10. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
11. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
12. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
13. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
14. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
15. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
16. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
17. Infinite Limits
Definition
The notation
y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking x
sufficiently close to a but not
equal to a.
. x
.
“Large” takes the place of
“close to L”.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
18. Negative Infinity
Definition
The notation
lim f(x) = −∞
x→a
means that the values of f(x) can be made arbitrarily large negative (as
large as we please) by taking x sufficiently close to a but not equal to a.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 8 / 38
19. Negative Infinity
Definition
The notation
lim f(x) = −∞
x→a
means that the values of f(x) can be made arbitrarily large negative (as
large as we please) by taking x sufficiently close to a but not equal to a.
We call a number large or small based on its absolute value. So
−1, 000, 000 is a large (negative) number.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 8 / 38
20. Vertical Asymptotes
Definition
The line x = a is called a vertical asymptote of the curve y = f(x) if at
least one of the following is true:
lim f(x) = ∞ lim f(x) = −∞
x→a x→a
lim f(x) = ∞ lim f(x) = −∞
x→a+ x→a+
lim f(x) = ∞ lim f(x) = −∞
x→a− x→a−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 9 / 38
21. Infinite Limits we Know
y
.
.
.
1
lim+ = ∞
x→0 x .
. . . . . . . x
.
.
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
22. Infinite Limits we Know
y
.
.
.
1
lim+ = ∞
x→0 x .
1
lim = −∞
x→0− x . . . . . . . x
.
.
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
23. Infinite Limits we Know
y
.
.
.
1
lim+ = ∞
x→0 x .
1
lim = −∞
x→0− x . . . . . . . x
.
1
lim =∞
x→0 x2 .
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
24. Finding limits at trouble spots
Example
Let
x2 + 2
f(x) =
x2 − 3x + 2
Find lim f(x) and lim+ f(x) for each a at which f is not continuous.
x→a− x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 11 / 38
25. Finding limits at trouble spots
Example
Let
x2 + 2
f(x) =
x2 − 3x + 2
Find lim f(x) and lim+ f(x) for each a at which f is not continuous.
x→a− x→a
Solution
The denominator factors as (x − 1)(x − 2). We can record the signs of
the factors on the number line.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 11 / 38
26. Use the number line
. . x − 1)
(
. . . . . . .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
27. Use the number line
−
.. 0
.. .
+
. x − 1)
.
.
(
1
.
. . . . . . .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
28. Use the number line
−
.. 0
.. .
+
. x − 1)
.
.
(
1
.
−
. 0
.. .
+
. x − 2)
.
.
(
2
.
. . . . . . .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
29. Use the number line
−
.. 0
.. .
+
. x − 1)
.
.
(
1
.
−
. 0
.. .
+
. x − 2)
.
.
(
2
.
.
+
. x2 + 2)
(
. . . . . . .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
30. Use the number line
−
.. 0
.. .
+
. x − 1)
.
.
(
1
.
−
. 0
.. .
+
. x − 2)
.
.
(
2
.
.
+
. x2 + 2)
(
. . .. . . .. .
f
.(x)
1
. 2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
31. Use the number line
−
.. 0
.. .
+
. x − 1)
.
.
(
1
.
−
. 0
.. .
+
. x − 2)
.
.
(
2
.
.
+
. x2 + 2)
(
.+ . .. . . .. .
f
.(x)
1
. 2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
32. Use the number line
. mall
−
.. 0
.. .
+
s
. x − 1)
.
(
1
.
−
. 0
.. .
+
. x − 2)
.
.
(
2
.
.
+
. x2 + 2)
(
.+ +∞ . .
. . . .. .
f
.(x)
1
. 2
.
lim f(x) = + ∞
x→1−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
33. Use the number line
. mall
−
.. 0
.. .
+
s
. x − 1)
.
(
1
.
−
. 0
.. .
+
. x − 2)
.
.
(
2
.
.
+
. x2 + 2)
(
.+ +∞ . . ∞ .
. − . .. .
f
.(x)
1
. 2
.
lim f(x) = + ∞
x→1−
lim f(x) = − ∞
x→1+
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
38. In English, now
To explain the limit, you can say:
“As x → 1− , the numerator approaches 3, and the denominator
approaches 0 while remaining positive. So the limit is +∞.”
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 13 / 38
39. The graph so far
lim f(x) = + ∞ lim f(x) = − ∞
x→1− x→2−
lim f(x) = − ∞ lim f(x) = + ∞
x→1+ x→2+
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
40. The graph so far
lim f(x) = + ∞ lim f(x) = − ∞
x→1− x→2−
lim f(x) = − ∞ lim f(x) = + ∞
x→1+ x→2+
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
41. The graph so far
lim f(x) = + ∞ lim f(x) = − ∞
x→1− x→2−
lim f(x) = − ∞ lim f(x) = + ∞
x→1+ x→2+
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
42. The graph so far
lim f(x) = + ∞ lim f(x) = − ∞
x→1− x→2−
lim f(x) = − ∞ lim f(x) = + ∞
x→1+ x→2+
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
43. The graph so far
lim f(x) = + ∞ lim f(x) = − ∞
x→1− x→2−
lim f(x) = − ∞ lim f(x) = + ∞
x→1+ x→2+
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
44. Limit Laws (?) with infinite limits
Fact
The sum of two positive or two negative infinite limits is infinite.
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
45. Rules of Thumb with infinite limits
∞
. +∞=∞
Fact
The sum of two positive or two negative infinite limits is infinite.
.
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
46. Rules of Thumb with infinite limits
∞
. +∞=∞
−
. ∞ + (−∞) = −∞
Fact
The sum of two positive or two negative infinite limits is infinite.
.
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞.
.x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
47. Rules of Thumb with infinite limits
∞
. +∞=∞
−
. ∞ + (−∞) = −∞
Fact
The sum of two positive or two negative infinite limits is infinite.
.
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞.
.x→a x→a x→a
Remark
We don’t say anything here about limits of the form ∞ − ∞.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
48. Rules of Thumb with infinite limits
Kids, don't try this at home!
Fact
The sum of a finite limit and an infinite limit is infinite.
If lim f(x) = L and lim g(x) = ±∞, then lim (f(x) + g(x)) = ±∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 16 / 38
49. Rules of Thumb with infinite limits
Kids, don't try this at home!
Fact
The sum of a finite limit and an infinite limit is infinite.
.
If lim f(x) = L and lim g(x) = ±∞, then lim (f(x) + g(x)) = ±∞.
x→a x→a x→a
L+∞=∞
.
L − ∞ = −∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 16 / 38
50. Rules of Thumb with infinite limits
Kids, don't try this at home!
Fact
The product of a finite limit and an infinite limit is infinite if the finite limit
is not 0.
If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞.
x→a x→a x→a
If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞.
x→a x→a x→a
If lim f(x) = L, lim g(x) = −∞, and L > 0, then
x→a x→a
lim f(x) · g(x) = −∞.
x→a
If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
51. Rules of Thumb with infinite limits
Kids, don't try this at home!
{
∞ if L > 0
. ·∞=
L
−∞ if L < 0.
Fact
The product of a finite limit and an infinite limit is infinite if the finite limit
is not 0. .
If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞.
x→a x→a x→a
If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞.
x→a x→a x→a
If lim f(x) = L, lim g(x) = −∞, and L > 0, then
x→a x→a
lim f(x) · g(x) = −∞.
x→a
If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
52. Rules of Thumb with infinite limits
Kids, don't try this at home!
{
∞ if L > 0
. ·∞=
L
−∞ if L < 0.
Fact
The product of a finite limit and an infinite limit is infinite if the finite limit
is not 0. .
If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞.
x→a x→a x→a
If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞.
x→a x→a { x→a
If lim f(x) =(−∞) = −∞−∞,Land0L > 0, then
. ·
L L, lim g(x) = if >
x→a x→a
lim f(x) · g(x) = −∞. ∞ if L < 0.
x→a
If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞.
x→a x→a x→a
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
53. Multiplying infinite limits
Kids, don't try this at home!
Fact
The product of two infinite limits is infinite.
If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞.
x→a x→a x→a
If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 18 / 38
54. Multiplying infinite limits
Kids, don't try this at home!
∞·∞=∞
. ∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
Fact
The product of two infinite limits is infinite. .
If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞.
x→a x→a x→a
If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞.
x→a x→a x→a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 18 / 38
55. Dividing by Infinity
Kids, don't try this at home!
Fact
The quotient of a finite limit by an infinite limit is zero.
f(x)
If lim f(x) = L and lim g(x) = ±∞, then lim = 0.
x→a x→a x→a g(x)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 19 / 38
56. Dividing by Infinity
Kids, don't try this at home!
Fact
The quotient of a finite limit by an infinite limit is zero.
f(x)
If lim f(x) = L and lim g(x) = ±∞, then lim = 0.
x→a x→a x→a g(x)
.
L
. =0
∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 19 / 38
57. 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.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 20 / 38
58. Indeterminate Limit forms
L
Limits of the form are indeterminate. There is no rule for evaluating
0
such a form; the limit must be examined more closely. Consider these:
1 −1
lim =∞ lim = −∞
x→0 x2 x→0 x2
1 1
lim+ = ∞ lim = −∞
x→0 x x→0− x
1 L
Worst, lim is of the form , but the limit does not exist, even
x→0 x sin(1/x) 0
in the left- or right-hand sense. There are infinitely many vertical
asymptotes arbitrarily close to 0!
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 21 / 38
59. Indeterminate Limit forms
Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate.
Example
1
The limit lim+ sin x · is of the form 0 · ∞, but the answer is 1.
x→0 x
1
The limit lim+ sin2 x · is of the form 0 · ∞, but the answer is 0.
x→0 x
1
The limit lim+ sin x · 2 is of the form 0 · ∞, but the answer is ∞.
x→0 x
Limits of indeterminate forms may or may not “exist.” It will depend on
the context.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 22 / 38
60. Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 23 / 38
61. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 24 / 38
62. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we like, by
taking x sufficiently large.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
63. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we like, by
taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve y = f(x)
if either
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
64. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we like, by
taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve y = f(x)
if either
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞
y = L is a horizontal line!
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
65. Basic limits at infinity
Theorem
Let n be a positive integer. Then
1
lim =0
x→∞ xn
1
lim =0
x→−∞ xn
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 26 / 38
66. Limit laws at infinity
Fact
Any limit law that concerns finite limits at a finite point a is still true if
the finite point is replaced by ±∞.
That is, if lim f(x) = L and lim g(x) = M, then
x→∞ x→∞
lim (f(x) + g(x)) = L + M
x→∞
lim (f(x) − g(x)) = L − M
x→∞
lim cf(x) = c · L (for any constant c)
x→∞
lim f(x) · g(x) = L · M
x→∞
f(x) L
lim = (if M ̸= 0), etc.
x→∞ g(x) M
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 27 / 38
67. Using the limit laws to compute limits at ∞
Example
x
Find lim
x→∞ x2 +1
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 28 / 38
68. Using the limit laws to compute limits at ∞
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0.
y
.
. x
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 28 / 38
69. Solution
Solution
Factor out the largest power of x from the numerator and denominator.
We have
x x(1) 1 1
= 2 = ·
x2
+1 x (1 + 1/x2 ) x 1 + 1/x2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2
1
=0· = 0.
1+0
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 29 / 38
70. Using the limit laws to compute limits at ∞
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0.
y
.
. x
.
Notice that the graph does cross the asymptote, which contradicts one
of the commonly held beliefs of what an asymptote is.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 30 / 38
71. Solution
Solution
Factor out the largest power of x from the numerator and denominator.
We have
x x(1) 1 1
= 2 = ·
x2
+1 x (1 + 1/x2 ) x 1 + 1/x2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2
1
=0· = 0.
1+0
Remark
Had the higher power been in the numerator, the limit would have been
∞.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 31 / 38
72. Another Example
Example
Find
2x3 + 3x + 1
lim
x→∞ 4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 32 / 38
73. Another Example
Example
Find
2x3 + 3x + 1
lim
x→∞ 4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 32 / 38
74. Solution
Solution
Factor out the largest power of x from the numerator and denominator.
We have
2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 )
= 3
4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 )
2x3 + 3x + 1 2 + 3/x2 + 1/x3
lim = lim
x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3
2+0+0 1
= =
4+0+0 2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 33 / 38
75. Solution
Solution
Factor out the largest power of x from the numerator and denominator.
We have
2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 )
= 3
4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 )
2x3 + 3x + 1 2 + 3/x2 + 1/x3
lim = lim
x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3
2+0+0 1
= =
4+0+0 2
Upshot
When finding limits of algebraic expressions at infinity, look at the
highest degree terms.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 33 / 38
76. Still Another Example
Example
Find √
3x4 + 7
lim
x→∞ x2 + 3
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 34 / 38
77. Still Another Example
√ √ √
. 3x4 + 7 ∼ 3x4 = 3x2
Example
Find √
3x4 + 7 .
lim
x→∞ x2 + 3
Answer
√
The limit is 3.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 34 / 38
79. Rationalizing to get a limit
Example
(√ )
Compute lim 4x2 + 17 − 2x .
x→∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 36 / 38
80. Rationalizing to get a limit
Example
(√ )
Compute lim 4x2 + 17 − 2x .
x→∞
Solution
This limit is of the form ∞ − ∞, which we cannot use. So we
rationalize the numerator (the denominator is 1) to get an expression
that we can use the limit laws on.
(√ ) (√ ) √4x2 + 17 + 2x
lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √
x→∞ x→∞ 4x2 + 17 + 2x
(4x2 + 17) − 4x2
= lim √
x→∞ 4x2 + 17 + 2x
17
= lim √ =0
x→∞ 4x2 + 17 + 2x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 36 / 38
81. Kick it up a notch
Example
(√ )
Compute lim 4x2 + 17x − 2x .
x→∞
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 37 / 38
83. Summary
Infinity is a more complicated concept than a single number.
There are rules of thumb, but there are also exceptions.
Take a two-pronged approach to limits involving infinity:
Look at the expression to guess the limit.
Use limit rules and algebra to verify it.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 38 / 38