lecture7 - is true ex Graph the given functions 1 f x = 1 x...

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Lecture 7: Limits (Section 2.2) Recall the de±nition: lim x ! a f ( x ) = L if we can make the values of f ( x ) as close to L as we want by choos- ing x su²ciently close to a on either side but not equal to a . ex. If f ( x ) = ± x if x 6 = 1 3 if x = 1 , ±nd lim x ! 1 f ( x ). 6 - ? ± ex. If g ( x ) = ± 3 if x ± 0 ² 1 if x > 0 , ±nd lim x ! 0 g ( x ). 6 - ? ±
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Def. One-Sided Limits lim x ! a ± f ( x ) lim x ! a + f ( x ) NOTE: lim x ! a f ( x ) = L if and only if In our example, lim x 1 ± g ( x ) = lim x 1 + g ( x ) =
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From the given graph, ±nd the limits: 6 - ? ± lim x 3 f ( x ) = lim x ! 2 ± f ( x ) = lim x 1 + f ( x ) = lim x ! 2 + f ( x ) = lim x 1 ± f ( x ) = lim x ! 2 f ( x ) = lim x 1 f ( x ) = lim x 2 f ( x ) =
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In±nite Limits Def. Let f be de±ned on both sides of a , except possibly at a . Then lim x ! a f ( x ) = 1 if the values of f ( x ) can be made as large as we want by taking x su²ciently close to a but not equal to a . lim x ! a f ( x ) = ±1 means the values of f ( x ) can be made as negatively large as we want by taking x su²ciently close but not equal to a . NOTE: Similar de±nitions can be given for ap- proaching a from the left or right. Def. The line x = a is called a of the curve y = f ( x ) if at least one of the following
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Unformatted text preview: is true: ex. Graph the given functions: 1) f ( x ) = 1 x ± 1 6-? ± 2) g ( x ) = 1 ( x ± 1) 2 6-? ± 3) h ( x ) = ± 1 ( x ± 1) 2 6-? ± Use the graphs to ±nd the following limits: 1) lim x ! 1 ± f ( x ) = lim x ! 1 + f ( x ) = lim x ! 1 f ( x ) = 2) lim x ! 1 g ( x ) = 3) lim x ! 1 h ( x ) = ex. Find the following limits: 1) lim x ! 2 + 2 2 ± x 2) lim x ! 1 ± x ± 5 x ± 1 ex. Find each vertical asymptote of f ( x ) = x ± 5 x ± 1 . ex. Find each vertical asymptote of f ( x ) = x 2 ± 1 x ± 1 (example, lecture 6). ex. Evaluate lim x ! 1 + ln( x ± 1). Find any vertical asymptotes of f ( x ) = ln( x ± 1). 6-? ± ex. Find lim x ! ± ± cot x . Find any vertical asymptotes of g ( x ) = cot x . 6-? ±...
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This note was uploaded on 02/10/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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lecture7 - is true ex Graph the given functions 1 f x = 1 x...

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