In this exercise we compare three functions: (a) Begin by

Chapter 5, Problem 25

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In this exercise we compare three functions: (a) Begin by graphing all three functions in the standard viewing rectangle. The picture confirms three facts that you know from the text: on the positive x-axis, the functions increase very rapidly; on the negative x-axis, the graphs approach the asymptote (which is the x-axis) as you move to the left; the y-intercept in each case is 1. (b) To compare the functions for positive values of x, use a viewing rectangle in which x extends from 0 to 3 and y extends from 0 to 10. Note that the graph of ex is y 2x y ex y 3x G G 1e G bounded between the graphs of 2x and 3x , just as the number e is between 2 and 3. In particular, the picture that you obtain demonstrates the following fact: For positive values of x,(c) To see the graphs more clearly when x is negative,change the viewing rectangle so that x extends from3 to 0 and y extends from 0 to 1. Again, note thatthe graph of e x is bounded between the graphs of 2xand 3x, but now the graph of 3x is the bottom (ratherthan the top) curve in the picture. This demonstratesthe following fact: For negative values of x,(d) Explain how the result in part (c) follows from the resultin part (b).