The document defines the limit of a function as x approaches a value a. Specifically: 1) The limit of f(x) as x approaches a, written as lim f(x) = L, means that the values of f(x) can be made arbitrarily close to L by making x sufficiently close to a, without x being equal to a. 2) For the limit to exist, the left and right hand limits must be equal. The function must also be bounded near a and cannot oscillate with increasingly high frequency. 3) Formally, the limit is L if for any positive epsilon, there corresponds a positive delta such that if the absolute value of x - a is less than

1. Section 2.2
The Limit of a Function
Math 1a
September 28, 2007

2. 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)

3. 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.

5. The error-tolerance game
L
a

6. The error-tolerance game
L
a

8. The error-tolerance game
L
a

9. The error-tolerance game
This tolerance is too big
L
a

10. The error-tolerance game
L
a

11. The error-tolerance game
Still too big
L
a

12. The error-tolerance game
L
a

13. The error-tolerance game
This looks good
L
a

14. The error-tolerance game
So does this
L
a

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

18. Examples
Example
Find lim x 2 if it exists.
x→0
Example
|x|
Find lim if it exists.
x→0 x

20. Examples
Example
Find lim x 2 if it exists.
x→0
Example
|x|
Find lim if it exists.
x→0 x
Example
1
Find lim+ if it exists.
x→0 x

22. Examples
Example
Find lim x 2 if it exists.
x→0
Example
|x|
Find lim if it exists.
x→0 x
Example
1
Find lim+ if it exists.
x→0 x
Example
π
Find lim sin if it exists.
x→0 x

24. 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

25. 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.

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

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

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

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

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

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

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