Chapter 4 | Applications of Derivatives
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Example 4.27 Determining End Behavior for a Transcendental Function
Find the limits as x →∞ and x →−∞ for f ( x ) = (2+3 e x ) (7−5 e x )
and describe the end behavior of f .
Solution To find the limit as x →∞, divide the numerator and denominator by e x : lim x →∞ f ( x ) = lim x →∞ 2+3 e x 7−5 e x
(2/ e x )+3 (7/ e x )−5 .
= lim x →∞
As shown in Figure 4.60 , e x →∞ as x →∞. Therefore, lim x →∞ 2 e x =0= lim x →∞ 7 e x . We conclude that lim x →∞ f ( x ) = − 3 5 , and the graph of f approaches the horizontal asymptote y = − 3 5 as x →∞. To find the limit as x →−∞, use the fact that e x →0 as x →−∞ to conclude that lim x →∞ f ( x ) = 2 7 , and therefore the graph of approaches the horizontal asymptote y = 2 7 as x →−∞.
e x −4) (5 e x +2) .
4.26
Find the limits as x →∞ and x →−∞ for f ( x ) = (3
Guidelines for Drawing the Graph of a Function We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.
Problem-Solving Strategy: Drawing the Graph of a Function Given a function f , use the following steps to sketch a graph of f : 1. Determine the domain of the function. 2. Locate the x - and y -intercepts. 3. Evaluate lim x →∞ f ( x ) and lim x →−∞
f ( x ) to determine the end behavior. If either of these limits is a finite number L , then y = L is a horizontal asymptote. If either of these limits is ∞ or −∞, determine whether f has an oblique asymptote. If f is a rational function such that f ( x ) = p ( x ) q ( x ) , where the degree of the numerator is greater than the degree of the denominator, then f can be written as
p ( x ) q ( x ) =
g ( x )+ r ( x )
f ( x ) =
q ( x ) ,
where the degree of r ( x ) is less than the degree of q ( x ). The values of f ( x ) approach the values of g ( x ) as
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