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Unformatted text preview: 2.5 Infinite Limits and Vertical Asymptotes
Infinite Limits
1 f (x) = x . What happens to the values of f (x) as x 0+ ? As x 0+ , the values of f grow without bound. Given any positive real number B, however large, the values of f become larger still. We say f (x) approaches as x 0+ (we are not saying that f has a (righthand) limit! ) and write As x 0 , the values of f (x) = We write 1 x become arbitrarily large and negative. 1 Example 2 limx0 1 x2 Example 1 Find 1 1 limx1+ x1 and limx1 x1 Example 3 (a) limx2 (x2) = x2 4 2 2 (f) 2x limx2 (x2)3 = Precise Definitions of Infinite Limits DEFINITIONS
1. We say that f (x) approaches infinity as x approaches x0 , and write lim f (x) = ,
xx0 if for every positive real number B there exists a corresponding > 0 such that for all x 0 < x  x0  < f (x) > B. 2. We say that f (x) approaches negative infinity as x approaches x0 , and write lim f (x) = ,
xx0 if for every negative real number B there exists a corresponding > 0 such that for all x 0 < x  x0  < f (x) < B. 3 Example 4 1 Prove that limx0 2 = . x Solution: Given B > 0, we want to find > 0 such that 4 Vertical Asymptotes
DEFINITION A line x = a is a vertical asymptote of the graph of a function y = f (x) if either Example 5 Find the horizontal and vertical asymptotes of the curve y= x+3 . x+2 5 Example 6 find the horizontal and vertical asymptotes of the graph of f (x) =  x2 8 . 4 Example 7 What is limx0+ ln x ? The graph of ln x is the reflection of the graph of ex across the line y = x. 6 Example 8 What are the vertical asymptotes of the following curves? sin 1 y = sec x = cos x and y = tan x = cos x . x 7 Example 9 Find the asymptotes of the graph of f (x) = x2  3 . 2x  4 8 ...
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This note was uploaded on 03/31/2008 for the course MATH 1205 taught by Professor Fbhinkelmann during the Spring '08 term at Virginia Tech.
 Spring '08
 FBHinkelmann
 Calculus, Asymptotes, Limits

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