The document discusses the concept of a limit in calculus. It provides an outline that covers heuristics, errors and tolerances, examples, pathologies, and a precise definition of a limit. It introduces the error-tolerance game used to determine if a limit exists and illustrates this through examples evaluating the limits of x^2 as x approaches 0 and |x|/x as x approaches 0. It also discusses one-sided limits and evaluates the limit of 1/x as x approaches 0 from the right.

1. Section 1.3
The Concept of Limit
V63.0121.006/016, Calculus I
January 26, 2009
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. . . . . .

2. Limit
. . . . . .

3. 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)
. . . . . .

4. Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .

5. 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.
. . . . . .

6. Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .

7. The error-tolerance game
A game between two players to decide if a limit lim f(x) exists.
x→a
Step 1 Player 1: Choose L to be the limit.
Step 2 Player 2: Propose an “error” level around L.
Step 3 Player 1: Choose a “tolerance” level around a so that
x-points within that tolerance level of a are taken to
y-values within the error level of L, with the possible
exception of a itself.
Step 4 Go back to Step 2 until Player 1 cannot move.
If Player 1 can always find a tolerance level, lim f(x) = L.
x →a
. . . . . .

8. The error-tolerance game
L
.
.
a
.
. . . . . .

9. The error-tolerance game
L
.
.
a
.
. . . . . .

10. 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.
. . . . . .

11. 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.
. . . . . .

12. 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.
. . . . . .

13. 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.
. . . . . .

14. 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.
. . . . . .

15. 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.
. . . . . .

16. 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.
. . . . . .

17. 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.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .

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.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .

19. Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .

20. Example
Find lim x2 if it exists.
x →0
. . . . . .

21. Example
Find lim x2 if it exists.
x →0
Solution
I claim the limit is zero.
. . . . . .

22. Example
Find lim x2 if it exists.
x →0
Solution
I claim the limit is zero.
If the error level is 0.01, I need to guarantee that
−0.01 < x2 < 0.01 for all x sufficiently close to zero.
. . . . . .

23. Example
Find lim x2 if it exists.
x →0
Solution
I claim the limit is zero.
If the error level is 0.01, I need 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 I win that round.
. . . . . .

24. Example
Find lim x2 if it exists.
x →0
Solution
I claim the limit is zero.
If the error level is 0.01, I need 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 I win that round.
What should the tolerance be if the error is 0.0001?
. . . . . .

25. Example
Find lim x2 if it exists.
x →0
Solution
I claim the limit is zero.
If the error level is 0.01, I need 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 I win that round.
What should the tolerance be if the error is 0.0001?
By setting tolerance equal to the square root of the error, we can
guarantee to be within any error.
. . . . . .

26. Example
|x|
Find lim if it exists.
x →0 x
. . . . . .

27. 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?
. . . . . .

28. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

29. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

30. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

31. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

32. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph in-
. 1.
− side blue is not
inside green
. . . . . .

33. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

34. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

35. The error-tolerance game
y
.
.
Part of graph in-
side blue is not . .
1
inside green
. x
.
. 1.
−
. . . . . .

36. The error-tolerance game
y
.
.
Part of graph in-
side blue is not . .
1
inside green
. x
.
. 1.
−
These are the only good choices; the limit does not exist.
. . . . . .

37. 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.
. . . . . .

38. 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.
. . . . . .

39. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

40. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

41. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

42. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

43. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph in-
. 1.
− side blue is in-
side green
. . . . . .

44. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

45. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

46. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .

47. The error-tolerance game
y
.
.
Part of graph in- . .
1
side blue is in-
side green
. x
.
. 1.
−
. . . . . .

48. The error-tolerance game
y
.
.
Part of graph in- . .
1
side blue is in-
side green
. x
.
. 1.
−
So lim f(x) = 1 and lim f(x) = −1
x→0+ x →0 −
. . . . . .

49. 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 −
. . . . . .

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

51. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

52. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

53. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

54. The error-tolerance game
y
.
.
The graph escapes the
green, so no good
.? .
L
. x
.
0
.
. . . . . .

55. The error-tolerance game
y
.
.? .
L
. x
.
0
.
. . . . . .

56. The error-tolerance game
y
.
E
. ven worse!
.? .
L
. x
.
0
.
. . . . . .

57. The error-tolerance game
y
.
.
The limit does not exist
because the function is
unbounded near 0
.? .
L
. x
.
0
.
. . . . . .

58. 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
. . . . . .

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

60. Function values
x π/x sin(π/x)
. /2
π
1 π 0 .
1 /2 2π 0
1/k kπ 0
2 π/2 1
2 /5 5π/2 1 . . . ..
π 0
2 /9 9π/2 1
2/13 13π/2 1
2 /3 3π/2 −1
2 /7 7π/2 −1 .
2/11 11π/2 −1 3
. π/2
. . . . . .

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

62. 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 =
. . . . . .

63. 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 =
. . . . . .

64. 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 =
. . . . . .

65. 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
. . . . . .

66. 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.
. . . . . .

67. Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .

68. 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
. . . . . .

69. 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
. . . . . .

70. Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
. . . . . .

71. 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.
. . . . . .

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

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

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

75. The error-tolerance game = ε, δ
T
. his δ is too big
L
. +ε
L
.
. −ε
L
.
aa
. − δ. . + δ
a
. . . . . .

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

77. The error-tolerance game = ε, δ
T
. his δ looks good
L
. +ε
L
.
. −ε
L
.
a . a+
. −. δ δ
a
. . . . . .

78. The error-tolerance game = ε, δ
S
. o does this δ
L
. +ε
L
.
. −ε
L
.
a
aa .+
. .− δ δ
. . . . . .