L15 - Lecture 15 Indeterminate Forms LHopitals Rule...

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Lecture 15 — Indeterminate Forms; L’Hˆopital’s Rule Indeterminate forms are limits for which we cannot plug in numbers immediately to evaluate nor can we reason the answer by inspection. These are NOT indeterminate forms: lim x →∞ arctan( x ) 1 + e - x lim x 0 x ln( x ) lim x →∞ 1 1 + x 2 lim x 1 1 1 - x 2 The indeterminate forms of interest to us are: 0 0 ( 0 · ∞ ) ( ) ( 1 ) ( 0 0 ) ( 0 ) For example, when looking at the quotient f ( x ) g ( x ) , suppose that when we try to use limit laws, we see that both f ( x ) 0 and g ( x ) 0 . The symbol 0 0 is not a real number, so the limit laws fail. In addition, we can not use the informal definitions to divine what will happen: if f ( x ) and g ( x ) are two numbers close to zero, their ratio could be anything! Thus, we need to manipulate the function by some other means in order to calculate the limit. Although we’ve seen many such techniques, we now learn an especially useful one called L’Hˆopital’s Rule.
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L’Hˆopital’s Rule Suppose that f and g are differentiable at a and g 0 ( x ) 6 = 0 in an open interval containing a (except possibly at a itself). Suppose that lim x a f ( x ) = 0 and lim x a g ( x ) = 0 or lim x a f ( x ) = ±∞ and lim x a g ( x ) = ±∞ Then lim x a f ( x ) g ( x ) = PROVIDED the second limit exists or is ±∞ .
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