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# lecture15 - Math 006(Lecture 15 Innite Limits Example 1 Let...

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Math 006 (Lecture 15) Infinite Limits Example 1. Let f ( x ) = 1 x - 1 , g ( x ) = 1 ( x - 1) 2 . Find lim x 1 f ( x ) and lim x 1 g ( x ). Definition 1. The vertical line x = a is a vertical asymptote for the graph of y = f ( x ) if f ( x ) → ∞ , or f ( x ) → -∞ , as x a + or x a - . Definition 2. f ( x ) is called a rational function if f ( x ) = n ( x ) d ( x ) where n ( x ) and d ( x ) are polynomials. Theorem 1. If f ( x ) = n ( x ) /d ( x ) is a rational function, d ( c ) = 0 but n ( c ) = 0 , then the line x = c is a vertical asymptote of the graph of f . 1

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Example 2. Find the vertical asymptote(s) of the graph of y = f ( x ). (a) f ( x ) = x 2 + x - 2 x 2 - 1 (b) f ( x ) = x - 3 x 2 - 4 x + 3 (c) f ( x ) = x 2 + 20 5( x - 2) 2 (d) f ( x ) = x - 1 ( x + 3) 2 Limits at Infinity Example 3. Consider f ( x ) = 1 x 2 and g ( x ) = x 2 . Find lim x →∞ f ( x ) and lim x →∞ g ( x ). Theorem 2. If p is a positive real number and k is any real constant, then 1. lim x →∞ k x p = 0 , lim x →-∞ k x p = 0 2. lim x →∞ kx p = ±∞ , lim x →-∞ kx p = ±∞ 3. If p ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 1 x + a 0 , a n = 0 , n 1 . then lim x →∞ p ( x ) = lim x →∞ a n x n = ±∞ , and lim x →-∞ p ( x ) = lim x →-∞ a n x n = ±∞ . Example 4. Let f ( x ) = 3 x 5 - 6 x . Find lim x →∞ f ( x ) and lim x →-∞ f ( x ). 2
Finding Horizontal Asymptotes Example 5. Find lim x →∞
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lecture15 - Math 006(Lecture 15 Innite Limits Example 1 Let...

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