Let f(x) (x5 1)/x2 . (a) Graph the function f using a

Chapter 4, Problem 53

(choose chapter or problem)

Let f(x) (x5 1)/x2 . (a) Graph the function f using a viewing rectangle that extends from 4 to 4 in the x-direction and from 8 to 8 in the y-direction. G 0 0 x3 2x2 1 x2 2x 1 x2 2x x G G x2 x 6 x 3 (b) Add the graph of the curve y x3 to your picture in part (a). Note that as x increases (that is, as x moves away from the origin), the graph of f looks more and more like the curve y x3 . For additional perspective, first change the viewing rectangle so that y extends from 20 to 20. (Retain the x-settings for the moment.) Describe what you see. Next, adjust the viewing rectangle so that x extends from 10 to 10 and y extends from 100 to 100. Summarize your observations. (c) In the text we said that a line is an asymptote for a curve if the distance between the line and the curve approaches zero as we move further and further out along the curve. The work in part (b) illustrates that a curve can behave like an asymptote for another curve. In particular, part (b) illustrates that the distance between the curve y x3 and the graph of the given function f approaches zero as we move further and further out along the graph of f. That is, the curve y x3 is an asymptote for the graph of the given function f. Complete the following two tables for a numerical perspective on this. In the tables, d denotes the vertical distance between the curve y x3 and the graph of f: x 5 10 50 100 500 d x 5 10 50 100 500 d (d) Parts (b) and (c) have provided both a graphical and a numerical perspective. For an algebraic perspective that ties together the previous results, verify the following identity, and then use it to explain why the results in parts (b) and (c) were inevitable: x5 1 x2 x3 1 x2 d ` x5 1 x2