This document is notes from a Calculus I class section on limits involving infinity. It discusses different types of infinite limits such as limits approaching positive or negative infinity. It defines vertical asymptotes and gives examples of common infinite limits. Rules for manipulating infinite limits are provided. The document also covers indeterminate forms where the limit is unclear and must be examined more closely. Finally, it discusses limits as x approaches infinity and defines horizontal asymptotes.
Lesson 12: Linear Approximation and Differentials (Section 21 handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Lesson 12: Linear Approximation and Differentials (Section 21 handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
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 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
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.
Intelligent analysis for historical macroseismic damage scenarios Fabrizio Gi...Beniamino Murgante
Intelligent analysis for historical macroseismic damage scenarios - Fabrizio Gizzi, Nicola Masini Maria Rosaria Potenza, Maria Danese, Cinzia Zotta - Archaeological and monumental heritage institute, National Research Council, Potenza (Italy),
Lucia Tilio, Beniamino Murgante - Laboratory of Urban and Territorial Systems, University of Basilicata (Italy)
Intelligent Analysis of Environmental Data (S4 ENVISA Workshop 2009)
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 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
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.
Intelligent analysis for historical macroseismic damage scenarios Fabrizio Gi...Beniamino Murgante
Intelligent analysis for historical macroseismic damage scenarios - Fabrizio Gizzi, Nicola Masini Maria Rosaria Potenza, Maria Danese, Cinzia Zotta - Archaeological and monumental heritage institute, National Research Council, Potenza (Italy),
Lucia Tilio, Beniamino Murgante - Laboratory of Urban and Territorial Systems, University of Basilicata (Italy)
Intelligent Analysis of Environmental Data (S4 ENVISA Workshop 2009)
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
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 24: Areas, Distances, the Integral (Section 021 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.
RSS stands for Really Simple Syndication or Rich Site Summary, depending on who you ask. Many websites publish a rich set of RSS feeds, which can be processed by other websites as a form of syndicated content. But the regular structure of RSS as an XML application means that feeds can be easily edited (“munged”) and combined (“mashed up”). Programming libraries exist for processing feeds, but Yahoo! Pipes makes this easy with a graphical user interface and no coding. We will discuss methods and applications of RSS feeds which might be suitable for a course website—for instance, combining feeds from SlideShare and scribd and publishing them to Facebook, or publishing your office hours on your blog automatically. (Received September 21, 2010)
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.
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.
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 handout)
1. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Notes
Section 1.6
Limits involving Infinity
V63.0121.021, Calculus I
New York University
September 22, 2010
Announcements
Quiz 1 is next week in recitation. Covers Sections 1.1–1.4
Announcements
Notes
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 22, 2010 2 / 36
Objectives
Notes
“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 22, 2010 3 / 36
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2. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Recall the definition of limit
Notes
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 22, 2010 4 / 36
Recall the unboundedness problem
Notes
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 22, 2010 5 / 36
Outline
Notes
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 22, 2010 6 / 36
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3. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Infinite Limits
Notes
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.
“Large” takes the place of x
“close to L”.
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 36
Negative Infinity
Notes
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 22, 2010 8 / 36
Vertical Asymptotes
Notes
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 22, 2010 9 / 36
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4. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Infinite Limits we Know
Notes
y
1
lim =∞
x→0+ x
1
lim = −∞
x→0− x x
1
lim =∞
x→0 x 2
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 10 / 36
Finding limits at trouble spots
Notes
Example
Let
x2 + 2
f (x) =
x 2 − 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 22, 2010 11 / 36
Use the number line
Notes
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
+ +∞ −∞ − −∞ +∞ +
f (x)
1 2
So
lim f (x) = +∞ lim f (x) = −∞
x→1− x→2−
lim f (x) = −∞ lim f (x) = +∞
x→1+ x→2+
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 36
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5. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
In English, now
Notes
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 22, 2010 13 / 36
The graph so far
Notes
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 22, 2010 14 / 36
Rules of Thumb with infinite limits
Notes
Fact ∞+∞=∞
If lim f (x) = ∞ and
x→a
lim g (x) = ∞, then
x→a
−∞ + (−∞) = −∞
lim (f (x) + g (x)) = ∞.
x→a
If lim f (x) = −∞ and
x→a
lim g (x) = −∞, then
x→a
lim (f (x) + g (x)) = −∞.
x→a
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 15 / 36
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6. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Rules of Thumb with infinite limits
Notes
L+∞=∞
L − ∞ = −∞
Fact
If lim f (x) = L and lim g (x) = ±∞, then
x→a x→a
lim (f (x) + g (x)) = ±∞.
x→a
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 16 / 36
Rules of Thumb with infinite limits
Kids, don’t try this at home! Notes
∞ 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 L > 0
L · (−∞) =
∞ if L < 0.
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 17 / 36
Multiplying infinite limits
Kids, don’t try this at home! Notes
∞·∞=∞
∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
Fact
The product of two infinite limits is infinite.
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 18 / 36
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7. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Dividing by Infinity
Kids, don’t try this at home! Notes
L
=0
∞
Fact
The quotient of a finite limit by an infinite limit is zero.
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 19 / 36
Dividing by zero is still not allowed
Notes
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 22, 2010 20 / 36
Indeterminate Limit forms
Notes
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→0x 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 22, 2010 21 / 36
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8. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Indeterminate Limit forms
Notes
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 22, 2010 22 / 36
Indeterminate forms are like Tug Of War
Notes
Which side wins depends on which side is stronger.
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 23 / 36
Outline
Notes
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 22, 2010 24 / 36
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9. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Definition Notes
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 22, 2010 25 / 36
Basic limits at infinity
Notes
Theorem
Let n be a positive integer. Then
1
lim =0
x→∞ x n
1
lim =0
x→−∞ x n
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 26 / 36
Limit laws at infinity
Notes
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 22, 2010 27 / 36
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10. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Using the limit laws to compute limits at ∞
Notes
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 22, 2010 28 / 36
Solution
Notes
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/x 2 ) x 1 + 1/x 2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x 2 + 1 x→∞ x 1 + 1/x 2 x→∞ x x→∞ 1 + 1/x 2
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 22, 2010 29 / 36
Another Example
Notes
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 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 22, 2010 30 / 36
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11. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Solution
Notes
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 31 / 36
Still Another Example
Notes
Example
Find √
3x 4 + 7
lim
x→∞ x2 + 3
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 32 / 36
Solution
Notes
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 33 / 36
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12. V63.0121.021, Calculus I Section 1.6 : Limits involving Infinity September 22, 2010
Rationalizing to get a limit
Notes
Example
Compute lim 4x 2 + 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.
√
4x 2 + 17 + 2x
lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √
x→∞ x→∞ 4x 2 + 17 + 2x
(4x 2 + 17) − 4x 2
= lim √
x→∞ 4x 2 + 17 + 2x
17
= lim √ =0
x→∞ 4x 2 + 17 + 2x
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 34 / 36
Kick it up a notch
Notes
Example
Compute lim 4x 2 + 17x − 2x .
x→∞
V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 35 / 36
Summary
Notes
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 22, 2010 36 / 36
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