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Unformatted text preview: (1/3/08) Section 1.4 Limits involving infinity Overview: In later chapters we will need notation and terminology to describe the behavior of functions in cases where the variable or the value of the function becomes large. We say that x or y tends to if it becomes an arbitrarily large positive number and that x or y tends to- if it becomes an arbitrarily large negative number. These concepts are the basis of the definitions of several types of limits that we discuss in this section. Topics: Infinite limits as x Finite limits as x One-sided and two-sided infinite limits Infinite limits of transcendental functions Infinite limits as x Imagine a point that moves on the curve y = x 3 in Figure 1. As the x-coordinate of the point increases through all positive values, the point moves to the right and rises higher and higher, so that it is eventually above any horizontal line, no matter how high it is. We say that x 3 tends to as x tends to and write lim x x 3 = . x 1 2 y 8 4 y = x 3 FIGURE 1 Similarly, as the x-coordinate of the point decreases through all negative values, the point moves to the left and drops lower and lower so that it is eventually beneath any horizontal line, regardless how low it is. We say that x 3 tends to- as x tends to- , and we write lim x - x 3 =- . The function y = x 3 illustrates the first and fourth parts of the following definition. Definition 1 (Infinite limits as x tends to ) (a) lim x f ( x ) = if f ( x ) is an arbitrarily large positive number for all sufficiently large positive x . (b) lim x f ( x ) =- if f ( x ) is an arbitrarily large negative number for all sufficiently large positive x . (c) lim x - f ( x ) = if f ( x ) is an arbitrarily large positive number for all sufficiently large negative x . (d) lim x - f ( x ) =- if f ( x ) is an arbitrarily large negative number for all sufficiently large negative x . When we say that a negative number x or y is large, we mean that its absolute value is large. 36 Section 1.4, Limits involving infinity p. 37 (1/3/08) Parts (a) and (b) of this definition apply only if f is defined on an interval ( a, ) for some number a , and parts (c) and (d) apply only if f is defined on (- , b ) for some b . We can often determine the types of limits described in Definition 1 from the graphs of the functions, as in the next example. Example 1 What are lim x x 2 and lim x - x 2 ? Solution The graph in Figure 2 shows that lim x x 2 = and lim x - x 2 = . square x 1 2- 2- 1 y 2 4 y = x 2 FIGURE 2 The next example illustrates a basic principle: any polynomial has the same limits as x and as x - as its term involving the highest power of x ....
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