Use the definition of limit to estimate limits. Use the definition of limit to estimate limits

Use the definition of limit to estimate limits.

• Use the definition of limit to estimate limits.

• Determine whether limits of functions exist.

• Use properties of limits and direct substitution to evaluate limits.

The notion of a limit is a fundamental concept of calculus.

• The notion of a limit is a fundamental concept of calculus.

• In this chapter, you will learn how to evaluate limits and how to use them in the two basic problems of calculus: the tangent line problem and the area problem.

Find the dimensions of a rectangle that has a perimeter of 24 inches and a maximum area.

• Find the dimensions of a rectangle that has a perimeter of 24 inches and a maximum area.

• Solution:

• Let w represent the width of the rectangle and let l represent the length of the rectangle. Because

• 2w + 2l = 24

• it follows that l = 12 – w, as shown in the figure.

So, the area of the rectangle is

• So, the area of the rectangle is

• A = lw

• = (12 – w)w

• = 12w – w2.

• Using this model for area, experiment with different values of w to see how to obtain the maximum area.

After trying several values, it appears that the maximum area occurs when w = 6, as shown in the table.

• After trying several values, it appears that the maximum area occurs when w = 6, as shown in the table.

• In limit terminology, you can say that “the limit of A as w approaches 6 is 36.” This is written as

Use a table to estimate numerically the limit: .

• Use a table to estimate numerically the limit: .

• Solution:

• Let f (x) = 3x – 2.

• Then construct a table that shows values of f (x) for two sets of x-values—one set that approaches 2 from the left and one that approaches 2 from the right.

From the table, it appears that the closer x gets to 2, the closer f (x) gets to 4. So, you can estimate the limit to be 4. Figure 12.1 illustrates this conclusion.

• From the table, it appears that the closer x gets to 2, the closer f (x) gets to 4. So, you can estimate the limit to be 4. Figure 12.1 illustrates this conclusion.

Next, you will examine some functions for which limits do not exist.

• Next, you will examine some functions for which limits do not exist.

Show that the limit does not exist.

• Show that the limit does not exist.

• Solution: Consider the graph of f (x) = | x |/x.

• From Figure 12.4, you can see that for positive x-values

• and for negative x-values

This means that no matter how close x gets to 0, there will be both positive and negative x-values that yield f (x) = 1 and f (x) = –1.

• This means that no matter how close x gets to 0, there will be both positive and negative x-values that yield f (x) = 1 and f (x) = –1.

• This implies that the limit does not exist.

Following are the three most common types of behavior associated with the nonexistence of a limit.

• Following are the three most common types of behavior associated with the nonexistence of a limit.

You have seen that sometimes the limit of f (x) as x  c is simply f (c), as shown in Example 2. In such cases, the limit can be evaluated by direct substitution.

• You have seen that sometimes the limit of f (x) as x  c is simply f (c), as shown in Example 2. In such cases, the limit can be evaluated by direct substitution.

• That is,

• There are many “well-behaved” functions, such as polynomial functions and rational functions with nonzero denominators, that have this property.

The following list includes some basic limits.

• The following list includes some basic limits.

• This list can also include trigonometric functions. For instance,

• and

By combining the basic limits with the following operations, you can find limits for a wide variety of functions.

• By combining the basic limits with the following operations, you can find limits for a wide variety of functions.

Find each limit.

• Find each limit.

• a. b. c.

• d. e. f.

• Solution:

• Use the properties of limits and direct substitution to evaluate each limit.

• a.

b.

• b.

• c.

d.

• d.

• e.

• f.

Example 9 shows algebraic solutions. To verify the limit in Example 9(a) numerically, for instance, create a table that shows values of x2 for two sets of x-values—one set that approaches 4 from the left and one that approaches 4 from the right, as shown below.

• Example 9 shows algebraic solutions. To verify the limit in Example 9(a) numerically, for instance, create a table that shows values of x2 for two sets of x-values—one set that approaches 4 from the left and one that approaches 4 from the right, as shown below.

From the table, you can see that the limit as x approaches 4 is 16. To verify the limit graphically, sketch the graph of y = x2. From the graph shown in Figure 12.7, you can determine that the limit as x approaches 4 is 16.

• From the table, you can see that the limit as x approaches 4 is 16. To verify the limit graphically, sketch the graph of y = x2. From the graph shown in Figure 12.7, you can determine that the limit as x approaches 4 is 16.

The following summarizes the results of using direct substitution to evaluate limits of polynomial and rational functions.

• The following summarizes the results of using direct substitution to evaluate limits of polynomial and rational functions.