The limit is the mathematical formulation of infinitesimal closeness.

1. Sec on 1.3
The Limit of a Func on
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University
January 31, 2011
Announcements
First wri en HW due Wednesday February 2
. Get-to-know-you survey and photo deadline is February 11

2. Announcements
First wri en HW due
Wednesday February 2
Get-to-know-you survey
and photo deadline is
February 11

3. Guidelines for written homework
Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (011), recita on number,
date, assignment number, book sec ons.
Explain your work and your reasoning in your own words. Use
complete English sentences.

4. Rubric
Points Descrip on of Work
3 Work is completely accurate and essen ally 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 explained, 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. Solu on is completely
incorrect.

5. Written homework: Don’t

6. Written homework: Do

7. Written homework: Do
Written Explanations

8. Written homework: Do
Graphs

9. Objectives
Understand and state the
informal defini on of a
limit.
Observe limits on a
graph.
Guess limits by algebraic
manipula on.
Guess limits by numerical
informa on.

10. Limit
.

11. Yoda on teaching course concepts
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.

12. Zeno’s Paradox
That which is in locomo on must
arrive at the half-way stage before
it arrives at the goal.
(Aristotle Physics VI:9, 239b10)

13. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit

14. Heuristic Definition of a Limit
Defini on
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.

15. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit

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.

17. 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.

18. 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 coun ng a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.

19. 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 coun ng 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.

20. The error-tolerance game
L
.
a

21. The error-tolerance game
L
.
a

22. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

23. The error-tolerance game
This tolerance is too big
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

24. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

25. The error-tolerance game
S ll too big
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

26. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

27. The error-tolerance game
This looks good
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

28. The error-tolerance game
So does this
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.

29. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can s ll move.

30. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can s ll move.

31. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit

32. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on

33. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana claims the limit is zero.

34. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
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.

35. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
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.

36. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?

37. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
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 .

38. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana has a shortcut: By se ng tolerance equal to the square
root of the error, Dana can win every round. Once Emerson
realizes this, Emerson must give up.

39. Graphical version of E-T game
with x2
y
.
x

40. Graphical version of E-T game
with x2
y
.
x

41. Graphical version of E-T game
with x2
y
.
x

42. Graphical version of E-T game
with x2
y
.
x

43. Graphical version of E-T game
with x2
y
.
x

44. Graphical version of E-T game
with x2
y
.
x

45. Graphical version of E-T game
with x2
y
.
x

46. Graphical version of E-T game
with x2
y
.
x

47. Graphical version of E-T game
with x2
y
No ma er how small an
error Emerson picks,
Dana can find a fi ng
tolerance band.
.
x

48. A piecewise-defined function
Example
|x|
Find lim if it exists.
x→0 x

49. A piecewise-defined function
Example
|x|
Find lim if it exists.
x→0 x
Solu on
The func on can also be wri en as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?

50. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
. x
−1

51. 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

52. 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

53. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
this for a tol- . x
erance?
−1

54. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
this for a tol- .
No. Part of x
erance? graph inside
−1 blue is not
inside green

55. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
Oh, I guess 1
the limit isn’t
1 .
No. Part of x
graph inside
−1 blue is not
inside green

56. 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

57. 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 . Can you fit xan
error of 0.5?
−1

58. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
. Can you fit xan
this for a tol-
error of 0.5?
erance?
−1

59. 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
How about inside green
this for a tol- . x
erance?
−1

60. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
No. Part of
graph inside
Oh, I guess 1 blue is not
the limit isn’t inside green
−1 . x
−1

61. 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

62. 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 . Can you fit xan
error of 0.5?
−1

63. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
. Can you fit xan
this for a tol-
error of 0.5?
erance?
−1

64. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
this for a tol- . No. None of x
erance? graph inside
−1 blue is inside
green

65. 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 . No. None of x
0 graph inside
−1 blue is inside
green

66. 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

67. One-sided limits
Defini on
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.

68. One-sided limits
Defini on
We write
lim f(x) = L
x→a−
and say
“the limit of f(x), as x approaches a from the le , 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.

69. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

70. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

71. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

72. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

73. 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

74. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

75. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

76. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
1
. x
−1

77. 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

78. 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

79. A piecewise-defined function
Example
|x|
Find lim if it exists.
x→0 x
Solu on
The error-tolerance game fails, but
lim f(x) = 1 lim f(x) = −1
x→0+ x→0−

80. Another Example
Example
1
Find lim+ if it exists.
x→0 x

81. The error-tolerance game with 1/x
y
1
Find lim+ if it exists. L?
x→0 x
. x
0

82. The error-tolerance game with 1/x
y
1
Find lim+ if it exists. L?
x→0 x
. x
0

83. The error-tolerance game with 1/x
y
1
Find lim+ if it exists. L?
x→0 x
. x
0

84. The error-tolerance game with 1/x
y
The graph escapes
the green, so no
good
1
Find lim+ if it exists. L?
x→0 x
. x
0

85. The error-tolerance game with 1/x
y
1
Find lim+ if it exists. L?
x→0 x
. x
0

86. The error-tolerance game with 1/x
y
Even worse!
1
Find lim+ if it exists. L?
x→0 x
. x
0

87. The error-tolerance game with 1/x
y
The limit does not exist
because the func on is
unbounded near 0
1
Find lim+ if it exists. L?
x→0 x
. x
0

88. Another Example
Example
1
Find lim+ if it exists.
x→0 x
Solu on
The limit does not exist because the func on is unbounded near 0.
Next week we will understand the statement that
1
lim+ = +∞
x→0 x

89. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x

90. Function values
x π/x sin(π/x)
π/2
π . 0
3π/2

91. Function values
x π/x sin(π/x)
π 0 π/2
π . 0
3π/2

92. Function values
x π/x sin(π/x)
π 0 π/2
2π 0
π . 0
3π/2

93. Function values
x π/x sin(π/x)
π 0 π/2
2π 0
kπ 0
π . 0
3π/2

94. Function values
x π/x sin(π/x)
1 π 0 π/2
2π 0
kπ 0
π . 0
3π/2

95. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
kπ 0
π . 0
3π/2

96. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π . 0
3π/2

97. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
π . 0
3π/2

98. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
5π/2 1 π . 0
3π/2

99. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
5π/2 1 π . 0
9π/2 1
3π/2

100. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
5π/2 1 π . 0
9π/2 1
((4k + 1)π)/2 1
3π/2

101. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
5π/2 1 π . 0
9π/2 1
((4k + 1)π)/2 1
3π/2

102. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
9π/2 1
((4k + 1)π)/2 1
3π/2

103. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
((4k + 1)π)/2 1
3π/2

104. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
3π/2

105. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
3π/2 −1
3π/2

106. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
3π/2 −1
7π/2 −1 3π/2

107. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
3π/2 −1
7π/2 −1 3π/2
((4k − 1)π)/2 −1

108. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
2/3 3π/2 −1
7π/2 −1 3π/2
((4k − 1)π)/2 −1

109. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
2/3 3π/2 −1
2/7 7π/2 −1 3π/2
((4k − 1)π)/2 −1

110. Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
2/(4k + 1) ((4k + 1)π)/2 1
2/3 3π/2 −1
2/7 7π/2 −1 3π/2
2/(4k − 1) ((4k − 1)π)/2 −1

111. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x

112. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
f(x) = 0 when x =
f(x) = 1 when x =
f(x) = −1 when x =

113. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =
f(x) = −1 when x =

114. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
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 =

115. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
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

116. Graph
Here is a graph of the func on:
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.

117. What could go wrong?
Summary of Limit Pathologies
How could a func on fail to have a limit? Some possibili es:
le - and right- hand limits exist but are not equal
The func on is unbounded near a
Oscilla on with increasingly high frequency near a

118. Meet the Mathematician
Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contribu ons in geometry,
calculus, complex analysis,
number theory
created the defini on of limit
we use today but didn’t
understand it

119. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit

120. Precise Definition of a Limit
No, this is not going to be on the test
Let f be a func on 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.

121. The error-tolerance game = ε, δ
L
.
a

122. The error-tolerance game = ε, δ
L+ε
L
L−ε
.
a

123. The error-tolerance game = ε, δ
L+ε
L
L−ε
.
a−δ a a+δ

124. The error-tolerance game = ε, δ
This δ is too big
L+ε
L
L−ε
.
a−δ a a+δ

125. The error-tolerance game = ε, δ
L+ε
L
L−ε
.
a−δ a a+δ

126. The error-tolerance game = ε, δ
This δ looks good
L+ε
L
L−ε
.
a−δ a a+δ

127. The error-tolerance game = ε, δ
So does this δ
L+ε
L
L−ε
.
a−δ aa+δ

128. Summary
Many perspectives on limits
Graphical: L is the value the func on
“wants to go to” near a y
Heuris cal: f(x) can be made arbitrarily 1
close to L by taking x sufficiently close
to a. . x
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a
−1
corresponding δ > 0 such that if
0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic: next me
FAIL