Stewart6e4_4 - MATH 191 Sections 1 and 3 Calculus I Fall...

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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 4.4: L’Hˆ opital’s Rule What if we want to evaluate the limit lim x → 1 ln( x ) x- 1 ? The problem here is that we have a limit of 0 in both the numerator and the denominator: the limit of the quotient could conceivably be just about anything. For example, if the numerator goes to 0 much more quickly than the denomi- nator, the limit could conceivably turn out to be . But if the denominator goes to 0 more quickly, then the limit of the quotient could even turn out to be ! Or perhaps the actual limit is somewhere in the middle. The “indeterminate” nature of the limit’s value leads us to call this an of type . We’ve seen limits like this one before, really: Examples. 1. We evaluated the limit lim t → 1 t 2- 1 t 3- 1 by factoring, as you should demon- strate here: 2. We used a different method, geometry, to evaluate the limit lim θ → sin( θ ) θ . What did this limit turn out to be? In the current limit, though, there’s nothing we can factor! We’ll see how to get around this in a moment. A different, but related, sort of problem arises in limits like lim...
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Stewart6e4_4 - MATH 191 Sections 1 and 3 Calculus I Fall...

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