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

2. Limit L a

3. Limits, Graphs, and Calculators

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

5. 3) Use your calculator to evaluate the limits
Answer : 16
Answer : no limit
Answer : no limit
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. This theorem is used to show a limit does not exist.
A Theorem
This theorem is used to show a limit does not exist.
For the function
But

11. Limit Theorems

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

18. More Examples

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

22. Examples
Find the limits

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

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

25. Examples

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

27. Asymptotes

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

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

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

33. Definition of continuity
The function f is continuous at the number c if
Defined at c

34. Types of Discontinuity
1- Removable discontinuity
f(a) is not defined or
Exist
f(a) is defined to
be a number other
than the value of
limit.

35. 2- Jump discontinuity

36. 3- Infinite discontinuity

37. Example Solution Discuss the continuity of f(x) at x = 2, where
Removable discontinuity

38. Example Solution Discuss the continuity of f(x) at x = 0, where
Jump discontinuity

39. Example Solution X=-3 Removable discontinuity X=0
Infinite discontinuity
X=5
Jump discontinuity
X=2
The function is continuous

40. Example
What value should b assigned to make the function continuous at x = 1
Solution

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

42. 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

43. 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.

44. 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

45. 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.