1. Limits and Continuity Definition Evaluation of Limits Continuity
Limits Involving Infinity

2. Limit L a

3. Limits, Graphs, and Calculators
c) Use direct substitution
**If direct substitution results in an indeterminate form, then try factoring or rationalizing to simplify f(x) and then try direct substitution again,
Indeterminate form

4. 2. Find 6
Note: f (-2) = 1
is not involved 2

5. 3) Use direct substitution to evaluate the limits
Answer : 16
Answer : DNE
Answer : DNE
Answer : 1/2

6. One-Sided Limit One-Sided Limits
The right-hand limit of f (x), as x approaches a, equals L
written:
if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a

7. The left-hand limit of f (x), as x approaches a, equals M
written:
if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. M a

8. Examples of One-Sided Limit
1. Given
Find
Find

9. More Examples
Find the limits:

10. A Theorem This theorem is used to show a limit does not exist.
Memorize
This theorem is used to show a limit does not exist.
For the function
But

11. Limit Theorems Page 99

12. Examples Using Limit Rule

13. More Examples

14. Indeterminate Forms
Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions.
Notice form
Ex.
Factor and cancel common factors

15. More Examples

16. The Squeezing Theorem
See Graph

17. Continuity
A function f is continuous at the point x = a if the following are true:
f(a) a

18. A function f is continuous at the point x = a if the following are true:
f(a)
Memorize a

19. Removable Discontinuities:
(You can fill the hole.)
Essential Discontinuities:
infinite
oscillating
jump

20. Examples Continuous everywhere except at
At which value(s) of x is the given function discontinuous?
Continuous everywhere
Continuous everywhere except at

21. F is continuous everywhere else h is continuous everywhere else
and
and
Thus F is not cont. at
Thus h is not cont. at x=1.
F is continuous everywhere else
h is continuous everywhere else

22. Continuous Functions If f and g are continuous at x = a, then
A polynomial function y = P(x) is continuous at every point x.
A rational function is continuous at every point x in its domain.

23. Intermediate Value Theorem
If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L.
Memorize
f (b)
f (c) = L
f (a) a c b

24. Example
f (x) is continuous (polynomial) and since f (1) < 0 and f (2) > 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0.

25. Limits at Infinity For all n > 0, provided that is defined.
Divide by
Ex.

26. More Examples

29. Infinite Limits
For all n > 0,
More Graphs

30. Examples
Find the limits

31. Limit and Trig Functions
From the graph of trigs functions
we conclude that they are continuous everywhere

32. Tangent and Secant
Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers

33. Inverse tangent function
f(x)=tan x is not one-to-one
But the function f(x)=tan x , -π/2 < x < π/2 is one-to-one. The restricted tangent function has an inverse function which is denoted by tan-1 or arctan and is called inverse tangent function.
Example: tan-1(1) = π/4 .
Limits involving tan-1:

34. Graphs of inverse functions
The graph of y = arctan x
Domain:
Range:

35. Limit and Exponential Functions
The above graph confirm that exponential functions are continuous everywhere.

36. Asymptotes

37. Examples
Find the asymptotes of the graphs of the functions