11.07 - Section 3.7: L'Hospital's Rule (concluded) Recall...

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Section 3.7: L’Hospital’s Rule (concluded) Recall the formal statement of L’Hospital’s Rule: Suppose lim x a f ( x ) = 0 and lim x a g ( x ) = 0. Suppose furthermore that f and g are differentiable and g ( x ) 0 near a except possibly at a itself. Then lim x a f ( x )/ g ( x ) = lim x a f ( x )/ g ( x ). The stipulation “ g ( x ) 0 near a except possibly at a itself” turns out to be necessary. There are crazy examples where g ( a ) 0, with g ( x ) oscillating wildly between positive and negative values in the vicinity of x = a , such that the limits lim x a f ( x )/ g ( x ) and lim x a f ( x )/ g ( x ) are “unequal” (in the sense that one exists and the other doesn’t). For instance, take f ( x ) = x sin(1/ x 4 ) exp(–1/ x 2 ) and g ( x ) = exp(–1/ x 2 ). Here is what f ( x )/ g ( x ) and f ( x )/ g ( x ) look like for x between 0 and 0.1: 0.02 0.04 0.06 0.08 0.10 - . 0 10 - . 0 05 . 0 05 . 0 10 0.02 0.04 0.06 0.08 0.10 - 150 - 100 - 50 50 100 150 The former approaches the limit 0 as x goes to 0, but the
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latter approaches no limit at all! So far we’ve looked at indeterminate expressions of type 0/0, i.e., limits of the form lim x a f ( x )/ g ( x ) with f ( x ), g ( x ) 0 as x a . Another type of indeterminate expression is typified by
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11.07 - Section 3.7: L'Hospital's Rule (concluded) Recall...

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