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.
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.
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.
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.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
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.
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 20: Derivatives and the Shapes of Curves (handout)
Lesson 6: Limits Involving ∞ (Section 21 slides)
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