This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
W001 - World Visions
Orario 09.30 – 13.00
Sala 4
TECHNOLOGIES, PARTNERSHIPS & BUSINESS MODELS
Developing new value-added services in the navigation markets
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.
W001 - World Visions
Orario 09.30 – 13.00
Sala 4
TECHNOLOGIES, PARTNERSHIPS & BUSINESS MODELS
Developing new value-added services in the navigation markets
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 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.
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.
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
V63.0121.041, Calculus I
New York University
September 13, 2010
Announcements
Let us know if you bought a WebAssign license last year and
cannot login
First written HW due Wednesday
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
Wednesday
Get-to-know-you survey
and photo deadline is
October 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 2 / 36
3. Guidelines for written homework
Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (041), recitation number, date,
assignment number, book sections.
Explain your work and your reasoning in your own words. Use
complete English sentences.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 3 / 36
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 4 / 36
5. Examples of written homework: Don't
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 5 / 36
6. Examples of written homework: Do
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 6 / 36
7. Examples of written homework: Do
Written Explanations
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 7 / 36
8. Examples of written homework: Do
Graphs
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 8 / 36
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 9 / 36
11. 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)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 10 / 36
13. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 12 / 36
15. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 14 / 36
16. The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
17. The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
18. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
19. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
20. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
21. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
22. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
23. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
24. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
25. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
28. Example
Find lim x2 if it exists.
x→0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
29. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
30. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
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.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
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.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
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? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
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 .
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
35. Example
|x|
Find lim if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 18 / 36
36. 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?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 18 / 36
37. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
38. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
39. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
40. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
41. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph
. 1.
− inside blue is not
inside green
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
42. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
43. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
44. The error-tolerance game
y
.
.
Part of graph
inside blue is not . .
1
inside green
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
45. The error-tolerance game
y
.
.
Part of graph
inside blue is not . .
1
inside green
. x
.
. 1.
−
These are the only good choices; the limit does not exist.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
46. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 20 / 36
47. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 20 / 36
48. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
49. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
50. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
51. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
52. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph
. 1.
− inside blue is
inside green
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
53. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
54. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
55. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
56. The error-tolerance game
y
.
.
Part of graph . .
1
inside blue is
inside green
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
57. The error-tolerance game
y
.
.
Part of graph . .
1
inside blue is
inside green
. x
.
. 1.
−
So lim+ f(x) = 1 and lim f(x) = −1
x→0 x→0−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
58. 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−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 22 / 36
59. Example
1
Find lim+ if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 23 / 36
60. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
61. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
62. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
63. The error-tolerance game
y
.
.
The graph escapes
the green, so no good
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
64. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
65. The error-tolerance game
y
.
E
. ven worse!
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
66. The error-tolerance game
y
.
.
The limit does not ex-
ist because the func-
tion is unbounded near
0
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
67. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 25 / 36
68. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 26 / 36
70. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
71. 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 =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
72. 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 =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
73. 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 =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
74. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
75. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 29 / 36
77. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 31 / 36
78. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 32 / 36
80. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 34 / 36
81. The error-tolerance game = ε, δ
L
.
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
82. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
83. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
84. The error-tolerance game = ε, δ
T
. his δ is too big
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
85. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
86. The error-tolerance game = ε, δ
T
. his δ looks good
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
87. The error-tolerance game = ε, δ
S
. o does this δ
L
. +ε
L
.
. −ε
L
.
. .− δ δ
aa .+
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
88. Summary: Many perspectives on limits
Graphical: L is the value the function “wants to go to” near a
Heuristical: f(x) can be made arbitrarily close to L by taking x
sufficiently close to a.
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic/Formulaic: next time
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 36 / 36