The concept of limit formalizes the notion of closeness of the function values to a certain value "near" a certain point. Limits behave well with respect to arithmetic--usually. Division by zero is always a problem, and we can't make conclusions about nonexistent limits!

1. Section 2.2–3
The Concept of Limit
Limit Laws
Math 1a
February 4, 2008
Announcements
Syllabus available on course website
Homework for Wednesday 2/6:
Practice 2.2: 1, 3, 5, 7, 13, 15; 2.3: 1, 3, 7, 13, 15, 17
Turn-in 2:2: 2, 4, 6, 8; 2.3: 2, 20, 38
Homework for Monday 2/11: 2.2.28, 2.3.30, 2.4.34
ALEKS due Wednesday 2/20

2. Outline
The Concept of Limit
Heuristics
Errors and tolerances
Pathologies
Limit Laws
Easy laws
The direct substitution property
Limits by algebra
Two more limit theorems

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. 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

7. The error-tolerance game
L
a

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

9. The error-tolerance game
L
a

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

11. The error-tolerance game
L
a

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

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

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

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

16. 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

17. 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

18. 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

19. 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.

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

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

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

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

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

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

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

27. 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

28. Outline
The Concept of Limit
Heuristics
Errors and tolerances
Pathologies
Limit Laws
Easy laws
The direct substitution property
Limits by algebra
Two more limit theorems

29. Limit Laws
Suppose that c is a constant and the limits
lim f (x) and lim g (x)
x→a x→a
exist. Then
1. lim [f (x) + g (x)] = lim f (x) + lim g (x)
x→a x→a x→a
2. lim [f (x) − g (x)] = lim f (x) − lim g (x)
x→a x→a x→a
3. lim [cf (x)] = c lim f (x)
x→a x→a
4. lim [f (x)g (x)] = lim f (x) · lim g (x)
x→a x→a x→a

30. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a

31. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x)
x→a x→a

32. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x) (follows from 3 repeatedly)
x→a x→a

33. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x) (follows from 3 repeatedly)
x→a x→a
7. lim c = c
x→a
8. lim x = a
x→a

34. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x) (follows from 3 repeatedly)
x→a x→a
7. lim c = c
x→a
8. lim x = a
x→a
9. lim x n = an
x→a
√ √
10. lim n x = n a
x→a

35. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x) (follows from 3 repeatedly)
x→a x→a
7. lim c = c
x→a
8. lim x = a
x→a
9. lim x n = an (follows from 6 and 8)
x→a
√ √
10. lim n x = n a
x→a

36. Limit Laws, continued
f (x) lim f (x)
5. lim = x→a , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x) (follows from 3 repeatedly)
x→a x→a
7. lim c = c
x→a
8. lim x = a
x→a
9. lim x n = an (follows from 6 and 8)
x→a
√ √
10. lim n x = n a
x→a
n
11. lim f (x) = n lim f (x) (If n is even, we must additionally
x→a x→a
assume that lim f (x) > 0)
x→a

37. Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain of
f , then
lim f (x) = f (a)
x→a

38. Limits do not see the point! (in a good way)
Theorem
If f (x) = g (x) when x = a, and lim g (x) = L, then lim f (x) = L.
x→a x→a

39. Limits do not see the point! (in a good way)
Theorem
If f (x) = g (x) when x = a, and lim g (x) = L, then lim f (x) = L.
x→a x→a
Example
x 2 + 2x + 1
Find lim , if it exists.
x→−1 x +1

40. Limits do not see the point! (in a good way)
Theorem
If f (x) = g (x) when x = a, and lim g (x) = L, then lim f (x) = L.
x→a x→a
Example
x 2 + 2x + 1
Find lim , if it exists.
x→−1 x +1
Solution
x 2 + 2x + 1
Since = x + 1 whenever x = −1, and since
x +1
x 2 + 2x + 1
lim x + 1 = 0, we have lim = 0.
x→−1 x→−1 x +1

41. Finding limits by algebraic manipulations
Example √
x −2
Find lim .
x→4 x −4

42. Finding limits by algebraic manipulations
Example √
x −2
Find lim .
x→4 x −4
Solution √ √ √
2
Write the denominator as x − 4 = x − 4 = ( x − 2)( x + 2).

43. Finding limits by algebraic manipulations
Example √
x −2
Find lim .
x→4 x −4
Solution √ 2 √ √
Write the denominator as x − 4 = x − 4 = ( x − 2)( x + 2).
So
√ √
x −2 x −2
lim = lim √ √
x→2 x − 4 x→2 ( x − 2)( x + 2)
1 1
= lim √ =
x→2 x +2 4

44. Finding limits by algebraic manipulations
Example √
x −2
Find lim .
x→4 x −4
Solution √ 2 √ √
Write the denominator as x − 4 = x − 4 = ( x − 2)( x + 2).
So
√ √
x −2 x −2
lim = lim √ √
x→2 x − 4 x→2 ( x − 2)( x + 2)
1 1
= lim √ =
x→2 x +2 4
Example√ √
3
x− 3a
Try lim .
x→a x −a

45. Two More Important Limit Theorems
Theorem
If f (x) ≤ g (x) when x is near a (except possibly at a), then
lim f (x) ≤ lim g (x)
x→a x→a
(as usual, provided these limits exist).
Theorem (The Squeeze/Sandwich/Pinching Theorem)
If f (x) ≤ g (x) ≤ h(x) when x is near a (as usual, except possibly
at a), and
lim f (x) = lim h(x) = L,
x→a x→a
then
lim g (x) = L.
x→a

46. We can use the Squeeze Theorem to make complicated limits
simple.

47. We can use the Squeeze Theorem to make complicated limits
simple.
Example
1
Show that lim x 2 sin = 0.
x→0 x

48. We can use the Squeeze Theorem to make complicated limits
simple.
Example
1
Show that lim x 2 sin = 0.
x→0 x
Solution
We have for all x,
1
−x 2 ≤ x 2 sin ≤ x2
x
The left and right sides go to zero as x → 0.