This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
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 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 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.
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 more good examples.
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.
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.
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 derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
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.
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 more good examples.
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.
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.
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 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.
Lesson 16: Inverse Trigonometric Functions (Section 021 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.
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 18: Maximum and Minimum Values (Section 021 slides)Mel Anthony Pepito
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
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 1.3
The Concept of Limit
V63.0121.021, Calculus I
New York University
.
September 14, 2010
Announcements
Let us know if you bought a WebAssign license last year and
cannot login
First written HW due Thursday
Get-to-know-you survey and photo deadline is October 1
. . . . . .
2. Announcements
Let us know if you bought
a WebAssign license last
year and cannot login
First written HW due
Thursday
Get-to-know-you survey
and photo deadline is
October 1
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 2 / 39
3. Guidelines for written homework
Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (021), recitation number, date,
assignment number, book sections.
Explain your work and your reasoning in your own words. Use
complete English sentences.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 3 / 39
4. Rubric
Points Description of Work
3 Work is completely accurate and essentially perfect.
Work is thoroughly developed, neat, and easy to read.
Complete sentences are used.
2 Work is good, but incompletely developed, hard to read,
unexplained, or jumbled. Answers which are not ex-
plained, even if correct, will generally receive 2 points.
Work contains “right idea” but is flawed.
1 Work is sketchy. There is some correct work, but most of
work is incorrect.
0 Work minimal or non-existent. Solution is completely in-
correct.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 4 / 39
5. Examples of written homework: Don't
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 5 / 39
6. Examples of written homework: Do
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 6 / 39
7. Examples of written homework: Do
Written Explanations
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 7 / 39
8. Examples of written homework: Do
Graphs
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 8 / 39
9. Objectives
Understand and state the
informal definition of a limit.
Observe limits on a graph.
Guess limits by algebraic
manipulation.
Guess limits by numerical
information.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 9 / 39
11. Yoda on teaching a concepts course
“You must unlearn what you have learned.”
In other words, we are building up concepts and allowing ourselves
only to speak in terms of what we personally have produced.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 11 / 39
12. Zeno's Paradox
That which is in
locomotion must arrive
at the half-way stage
before it arrives at the
goal.
(Aristotle Physics VI:9, 239b10)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 12 / 39
14. Heuristic Definition of a 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 1.3 The Concept of Limit September 14, 2010 14 / 39
16. The error-tolerance game
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
Step 2 Emerson challenges with an “error” level around L.
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not counting a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.
Step 4 If Dana’s move is a good one, Emerson can challenge again or
give up. If Emerson gives up, Dana wins and the limit is L.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 16 / 39
17. The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
18. The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
19. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
20. The error-tolerance game
T
. his tolerance is too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
21. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
22. The error-tolerance game
S
. till too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
23. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
24. The error-tolerance game
T
. his looks good
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
25. The error-tolerance game
S
. o does this
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
26. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can still move.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
27. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can still move.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 17 / 39
29. Example
Find lim x2 if it exists.
x→0
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
30. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
31. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
32. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
33. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
34. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
35. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .
Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizes
this, Emerson must give up.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 19 / 39
36. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
37. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
38. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
39. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
40. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
41. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
42. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
43. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
44. Graphical version of the E-T game with x2
. .
y
. . .
x
.
.
No matter how small an error Emerson picks, Dana can find a
fitting tolerance band.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 20 / 39
45. Example
|x|
Find lim if it exists.
x→0 x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 21 / 39
46. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 21 / 39
47. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
48. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
I
. think the limit is 1
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
49. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
I
. think the limit is 1
. . ..
x
C
. an you fit an error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
50. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
. How about this . ..
x
for a tolerance?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
51. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
. How about this . ..
x
for a tolerance?
.
No. Part of
graph inside
. 1.
−
blue is not inside
green
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
52. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
Oh, I guess the
limit isn’t 1
. . ..
x
.
No. Part of
graph inside
. 1.
−
blue is not inside
green
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
53. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
I think the limit is
−1
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
54. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
I think the limit is
−1
. . . ..
x
Can you fit an
error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
55. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
. How about this . . ..
Can you fit an x
for a tolerance?
error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
56. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
.
No. Part of
. graph inside
. .
1
blue is not inside
green
.
. How about this . ..
x
for a tolerance?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
57. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
.
No. Part of
. graph inside
. .
1
. blue is not inside
Oh, I guess the green
limit isn’t −1
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
58. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
I think the limit is 0
. . ..
x
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
59. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
I think the limit is 0
. . . ..
x
Can you fit an
error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
60. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
. How about this . . ..
Can you fit an x
for a tolerance?
error of 0.5?
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
61. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
. How about this . ..
x
for a tolerance?
.
No. None of
. 1.
−
graph inside blue
is inside green
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
62. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
. Oh, I guess the . ..
x
limit isn’t 0
.
No. None of
. 1.
−
graph inside blue
is inside green
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
63. The E-T game with a piecewise function
|x|
Find lim if it exists.
x→0 x
y
.
.
. . .
1
.
I give up! I
. guess there’s . ..
x
no limit!
. 1.
−
.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 22 / 39
64. One-sided limits
Definition
We write
lim f(x) = L
x→a+
and say
“the limit of f(x), as x approaches a from the right, 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 and greater than a.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 23 / 39
65. One-sided limits
Definition
We write
lim f(x) = L
x→a−
and say
“the limit of f(x), as x approaches a from the left, 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 and less than a.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 23 / 39
66. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
67. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
68. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
69. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
70. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
.
Part of graph
. 1.
− inside blue is
inside green
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
71. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
72. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
73. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
74. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
.
Part of graph . .
1
inside blue is
inside green
. x
.
. 1.
−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 24 / 39
75. The error-tolerance game
|x| |x|
Find lim+ and lim if they exist.
x→0 x x→0 − x
y
.
.
Part of graph . .
1
inside blue is
inside green
. x
.
. 1.
−
So lim f(x) = 1 and lim f(x) = −1 . . . . . .
x→0+
V63.0121.021, Calculus I (NYU) x→0− 1.3
Section The Concept of Limit September 14, 2010 24 / 39
76. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
The error-tolerance game fails, but
lim f(x) = 1 lim f(x) = −1
x→0+ x→0−
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 25 / 39
77. Example
1
Find lim+ if it exists.
x→0 x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 26 / 39
78. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
79. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
80. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
81. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
.
The graph escapes
the green, so no good
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
82. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
83. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
E
. ven worse!
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
84. The error-tolerance game
1
Find lim+ if it exists.
x→0 x
y
.
.
The limit does not exist be-
cause the function is un-
bounded near 0
.?.
L
. x
.
0
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 27 / 39
85. Example
1
Find lim+ if it exists.
x→0 x
Solution
The limit does not exist because the function is unbounded near 0.
Next week we will understand the statement that
1
lim+ = +∞
x→0 x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 28 / 39
86. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 29 / 39
88. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
89. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
f(x) = 0 when x =
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
90. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
91. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
f(x) = −1 when x =
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
92. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
2
f(x) = −1 when x = for any integer k
4k − 1
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 31 / 39
93. Weird, wild stuff continued
Here is a graph of the function:
y
.
. .
1
. x
.
. 1.
−
There are infinitely many points arbitrarily close to zero where f(x) is 0,
or 1, or −1. So the limit cannot exist.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 32 / 39
95. What could go wrong?
Summary of Limit Pathologies
How could a function fail to have a limit? Some possibilities:
left- and right- hand limits exist but are not equal
The function is unbounded near a
Oscillation with increasingly high frequency near a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 34 / 39
96. Meet the Mathematician: Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contributions in
geometry, calculus,
complex analysis, number
theory
created the definition of
limit we use today but
didn’t understand it
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 35 / 39
98. Precise Definition of a Limit
No, this is not going to be on the test
Let f be a function defined on an some open interval that contains the
number a, except possibly at a itself. Then we say that the limit of f(x)
as x approaches a is L, and we write
lim f(x) = L,
x→a
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f(x) − L| < ε.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 37 / 39
99. The error-tolerance game = ε, δ
L
.
.
a
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
100. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
a
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
101. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
102. The error-tolerance game = ε, δ
T
. his δ is too big
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
103. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
104. The error-tolerance game = ε, δ
T
. his δ looks good
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
105. The error-tolerance game = ε, δ
S
. o does this δ
L
. +ε
L
.
. −ε
L
.
. .− δ δ
aa .+
a
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 38 / 39
106. Summary: Many perspectives on limits
Graphical: L is the value
the function “wants to go
to” near a
y
.
Heuristical: f(x) can be
made arbitrarily close to L . .
1
by taking x sufficiently
close to a.
. x
.
Informal: the
error/tolerance game
Precise: if for every ε > 0
. 1.
−
there is a corresponding
δ > 0 such that if
0 < |x − a| < δ, then
|f(x) − L| < ε.
FAIL
.
Algebraic: next time
. . . . . .
V63.0121.021, Calculus I (NYU) Section 1.3 The Concept of Limit September 14, 2010 39 / 39