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

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 tarix 20.09.2018 ölçüsü 1,5 Mb. #70013  • ## Use properties of limits and direct substitution to evaluate limits.  • ## 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. • ## it follows that l = 12 – w, as shown in the figure. • ## Using this model for area, experiment with different values of w to see how to obtain the maximum area. • ## In limit terminology, you can say that “the limit of A as w approaches 6 is 36.” This is written as   • ## 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.  • ## Next, you will examine some functions for which limits do not exist. • ## and for negative x-values • ## This implies that the limit does not exist. • ## Following are the three most common types of behavior associated with the nonexistence of a limit.  • ## There are many “well-behaved” functions, such as polynomial functions and rational functions with nonzero denominators, that have this property. • ## and • ## By combining the basic limits with the following operations, you can find limits for a wide variety of functions. • ## a. • ## c. • ## 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. • ## 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. Yüklə 1,5 Mb.

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