Chapter 4 | Applications of Derivatives
409
Figure 4.41 (a) As x →∞, the values of f are getting arbitrarily close to L . The line y = L is a horizontal asymptote of f . (b) As x →−∞, the values of f are getting arbitrarily close to M . The line y = M is a horizontal asymptote of f .
A function cannot cross a vertical asymptote because the graph must approach infinity (or −∞) from at least one direction as x approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function f ( x ) = (cos x ) x +1 shown in Figure 4.42 intersects the horizontal asymptote y =1 an infinite number of times as it oscillates around the asymptote with ever- decreasing amplitude.
Figure 4.42 The graph of f ( x ) = (cos x )/ x +1 crosses its horizontal asymptote y =1 an infinite number of times.
The algebraic limit laws and squeeze theorem we introduced in Introduction to Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity. Example 4.21 Computing Limits at Infinity
f ( x ) and lim x
f ( x ). Determine the horizontal
For each of the following functions f , evaluate lim x →∞
→−∞
asymptote(s) for f .
a. f ( x ) =5− 2 x 2 b. f ( x ) = sin x x c. f ( x ) = tan −1 ( x )
Made with FlippingBook - professional solution for displaying marketing and sales documents online