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

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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. 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 xvalues—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 xvalues and for negative xvalues
This means that no matter how close x gets to 0, there will be both positive and negative xvalues 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 xvalues 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. That is, There are many “wellbehaved” 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.
d.
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 xvalues—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 xvalues—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.
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