LIMITS OF FUNCTIONS

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.  

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

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

This leads us to the following general idea. 1.1.1 (p. 70) Limits (An Informal View)

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.

Figure 1.1.9 (p. 71) x .99 .999 .9999 .99999 1 1.00001 1.0001 1.001 1.01 F(x) 1.9949 1.9995 1.99995 1.999995 2.000005 2.00005 2.0005 2.004915

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.

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

Figure 1.2.1 (p. 80)

Example 1. Example 2.

The following theorem will be our basic tool for finding limits algebraically. Theorem 1.2.2 (p. 81)

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.

EXAMPLE : Evaluate the following limits.

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:

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

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.

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

Solution: (indeterminate) By Factoring:

Solution:

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.

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

This leads to the general idea of a one-sided limit 1.1.2 (p. 72) One-Sided Limits (An Informal View)

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

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

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.

Figure 1.1.13 (p. 73)

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

y 4 x -6 -2 4

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

EXAMPLE: SOLUTION

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 ,

x x

In general, we will use the following notation. 1.3.1 (p. 89) Limits at Infinity (An Informal View)

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

Fig.1.3.2 illustrates the graph of . As suggested by this graph, EXAMPLE Fig.1.3.2 illustrates the graph of . As suggested by this graph, Figure 1.3.4 (p. 90)

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

EXERCISES: A. Evaluate the following limits.

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

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