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

2. Limit L a

3. Limits, Graphs, and Calculators

4. Graph 3

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

6. 3) Use your calculator to evaluate the limits
Answer : 16
Answer : no limit
Answer : no limit
Answer : 1/2

7. The Definition of Limita

9. Examples
What do we do with the x?

10. 1/2 1
3/2

11. 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

12. 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

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

14. More Examples
Find the limits:

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

16. Limit Theorems

17. Examples Using Limit Rule

18. More Examples

19. 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

20. More Examples

21. The Squeezing Theorem
See Graph

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

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

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

25. 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

26. 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.

27. 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.
f (b)
f (c) = L
f (a) a c b

28. 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.

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

30. More Examples

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

34. Examples
Find the limits

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

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

37. Examples

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

39. Asymptotes

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