The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.

1. Section 1.3
The Limit of a Function
V63.0121, Calculus I
January 26–27, 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
The Concept of Limit
Heuristics
Errors and tolerances
Examples
Pathologies

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. The error-tolerance game
A game between two players to decide if a limit lim f (x) exists.
x→a
Player 1: Choose L to be the limit.
Player 2: Propose an “error” level around L.
Player 1: Choose a “tolerance” level around a so that x-points
within that tolerance level are taken to y -values within the
error level.
If Player 1 can always win, lim f (x) = L.
x→a

7. The error-tolerance game
L
a

8. The error-tolerance game
L
a

9. 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.

10. The error-tolerance game
This 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.

11. 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.

12. The error-tolerance game
Still 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.

13. 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.

14. The error-tolerance game
This 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.

15. The error-tolerance game
So 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.

16. 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.

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. Example
Find lim x 2 if it exists.
x→0

19. Example
Find lim x 2 if it exists.
x→0
Solution
By setting tolerance equal to the square root of the error, we can
guarantee to be within any error.

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

21. 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?

22. The error-tolerance game
y
1
x
−1

23. The error-tolerance game
y
1
x
−1

24. The error-tolerance game
y
1
x
−1

25. The error-tolerance game
y
1
x
−1

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

27. The error-tolerance game
y
1
x
−1

28. The error-tolerance game
y
1
x
−1

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

30. 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.

31. 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 (on either side
of a) and greater than a.

32. 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 (on either side
of a) and less than a.

33. 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+

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

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

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

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

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

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

40. The error-tolerance game
y
Even worse!
L?
x
0

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

42. Example
1
Find lim+ if it exists.
x
x→0
Solution
The limit does not exist because the function is unbounded near 0.
Next week we will understand the statement that
1
lim+ = +∞
x
x→0

43. Example
π
Find lim sin if it exists.
x
x→0

44. y
1
x
−1

45. What could go wrong?
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

46. 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

47. Precise Definition of a Limit
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.

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

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

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

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

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

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

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