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2311lectureoutline2.6 - x f x L →-∞ = Theorem If r> 0...

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MAC2311, Calculus I 2.6 Limits at Infinity: Horizontal Asymptotes Definition: Let f(x) be a function defined on some interval ( ) , a . Then ( ) lim x f x L →∞ = means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition: Let f(x) be a function defined on some interval ( ) , a -∞ . Then ( ) lim x f x L →-∞ = means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently small (large negative). Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either ( ) lim x f x L →∞ = or ( ) lim x f x L →-∞
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Unformatted text preview: x f x L →-∞ = . Theorem: If r > 0 is a rational number, then 1 lim r x x →∞ = . If r > 0 is a rational number such that x r is defined for all x , then 1 lim r x x →-∞ = . In General: • If a function f(x) approaches a certain number L as x gets larger and larger (or smaller and smaller), then f(x) has a horizontal asymptote at y = L . • A function can have zero, one or two horizontal asymptotes. • The graph of f(x) may intersect a horizontal asymptote, but never a vertical asymptote. Try Exercises 2, 4, 6, 13, 20, 22, 30, 40, 48 on pages 140-142 in the textbook....
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