1. LIMITS OF FUNCTIONS

2. LIMITS OF FUNCTIONS OBJECTIVES: define limits;
illustrate limits and its theorems; and
evaluate limits applying the given theorems.
define one-sided limits
illustrate one-sided limits
investigate the limit if it exist or not using the concept of one-sided limits.
define limits at infinity;
illustrate the limits at infinity; and
determine the horizontal asymptote.

3. DEFINITION: LIMITS The most basic use of limits is to describe how a
function behaves as the independent variable
approaches a given value. For example let us examine the behavior of the function
for x-values closer and closer to 2. It is evident from the graph and the table in the next slide that the values of f(x) get closer and closer to 3 as the values of x are selected closer and closer to 2 on either the left or right side of 2. We describe this by saying that the “limit of is 3 as x approaches 2 from either side,” we write

4. y
f(x) 3
f(x) x O 2 x
1.9
1.95
1.99
1.995
1.999 2
2.001
2.005
2.01
2.05
2.1
F(x)
2.71
2.852
2.97
2.985
2.997
3.003
3.015
3.031
3.152
3.31
left side
right side

5. This leads us to the following general idea.
(p. 70) Limits (An Informal View)

6. EXAMPLE
Use numerical evidence to make a conjecture about the value of
Although the function is undefined at x=1, this has no bearing on the limit.
The table shows sample x-values approaching 1 from the left side and from the right side. In both cases the corresponding values of f(x) appear to get closer and closer to 2, and hence we conjecture that
and is consistent with the graph of f.

7. Figure (p. 71) x
.99
.999
.9999
.99999 1
1.0001
1.001
1.01
F(x)
1.9949
1.9995
2.0005

8. THEOREMS ON LIMITS
Our strategy for finding limits algebraically has two parts:
First we will obtain the limits of some simpler function
Then we will develop a list of theorems that will enable us to use the limits of simple functions as building blocks for finding limits of more complicated functions.

9. We start with the following basic theorems, which are illustrated in Fig 1.2.1
Theorem (p. 80)

10. Figure (p. 80)

11. Example 1.
Example 2.

12. The following theorem will be our basic tool for finding limits algebraically.
Theorem (p. 81)

13. This theorem can be stated informally as follows:
The limit of a sum is the sum of the limits.
The limit of a difference is the difference of the limits.
The limits of a product is the product of the limits.
The limits of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.
The limit of the nth root is the nth root of the limit.
A constant factor can be moved through a limit symbol.

14. EXAMPLE :
Evaluate the following limits.

16. OR
When evaluating the limit of a function at a given value, simply replace the variable by the indicated limit then solve for the value of the function:

17. Evaluate the following limits.
EXAMPLE:
Evaluate the following limits.
Solution:
(indeterminate)
Equivalent function:

18. Note: In evaluating a limit of a quotient which reduces to , simplify the fraction. Just remove
the common factor in the numerator and denominator which makes the quotient .
To do this use factoring or rationalizing the numerator or denominator, wherever the radical is.

19. Rationalizing the numerator:
Solution:
(indeterminate)
Rationalizing the numerator:

20. Solution:
(indeterminate)
By Factoring:

21. Solution:

22. DEFINITION: One-Sided Limits
The limit of a function is called two-sided limit if it requires the values of f(x) to get closer and closer to a number as the values of x are taken from either side of x=a. However some functions exhibit different behaviors on the two sides of an x-value a in which case it is necessary to distinguish whether the values of x near a are on the left side or on the right side of a for purposes of investigating limiting behavior.

23. Consider the function As x approaches 0 from the right, the
values of f(x) approach a limit of 1, and
similarly , as x approaches 0 from the
left, the values of f(x) approach a
limit of -1. 1 -1

24. This leads to the general idea of a one-sided limit
(p. 72) One-Sided Limits (An Informal View)

25. 1.1.3 (p. 73) The Relationship Between One-Sided and Two-Sided Limits

26. EXAMPLE:
1. Find if the two sided limits exist given
SOLUTION 1 -1

27. EXAMPLE: 2. For the functions in Fig 1.1.13, find the one-sided
limit and the two-sided limits at x=a if they exists.
SOLUTION
The functions in all three figures have the same
one-sided limits as , since the functions are
Identical, except at x=a.
In all three cases the two-sided limit does not exist as
because the one sided limits are not equal.

28. Figure (p. 73)

29. EXAMPLE: 3. Find if the two-sided limit exists and sketch the graph of
SOLUTION

30. y 4 x -6 -2 4

31. EXAMPLE: 4. Find if the two-sided limit exists and sketch the graph
of and sketch the graph.
SOLUTION

32. EXAMPLE:
SOLUTION

34. DEFINITION: LIMITS AT INFINITY
If the values of the variable x increase without bound, then we write , and if the values of x decrease without bound, then we write
The behavior of a function as x increases or decreases without bound is sometimes called the end behavior of the function.
For example ,

35. x x

36. In general, we will use the following notation.
(p. 89) Limits at Infinity (An Informal View)

37. Fig.1.3.2 illustrates the end behavior of the function f when
Figure (p. 89)

38. Fig.1.3.2 illustrates the graph of . As suggested by this graph,
EXAMPLE
Fig illustrates the graph of As suggested by
this graph,
Figure (p. 90)

39. EXAMPLE ( Examples 7-11 from pages 92-95)

40. EXERCISES:
A. Evaluate the following limits.

41. EXERCISES:
B. Sketch the graph of the following functions and the indicated limit if it exists. find .

42. .