Chapter 4 | Applications of Derivatives
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4.6 | Limits at Infinity and Asymptotes Learning Objectives 4.6.1 Calculate the limit of a function as x increases or decreases without bound. 4.6.2 Recognize a horizontal asymptote on the graph of a function. 4.6.3 Estimate the end behavior of a function as x increases or decreases without bound.
4.6.4 Recognize an oblique asymptote on the graph of a function. 4.6.5 Analyze a function and its derivatives to draw its graph.
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x →±∞. In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f . Limits at Infinity We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs , we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes. Limits at Infinity and Horizontal Asymptotes Recall that lim x → a f ( x ) = L means f ( x ) becomes arbitrarily close to L as long as x is sufficiently close to a . We can extend this idea to limits at infinity. For example, consider the function f ( x ) =2+ 1 x . As can be seen graphically in Figure 4.40 and numerically in Table4.2 , as the values of x get larger, the values of f ( x ) approach 2. We say the limit as x approaches ∞ of f ( x ) is 2 andwrite lim x →∞ f ( x ) =2. Similarly, for x <0, as the values | x | get larger, the values of f ( x ) approaches 2. We say the limit as x approaches −∞ of f ( x ) is 2 and write lim x → -∞ f ( x ) =2.
Figure 4.40 The function approaches the asymptote y =2 as x approaches ±∞.
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