Unformatted text preview: MTH 174, section 4.4 Recall that, given , the tangent line to the graph at has equation ^ Thm L'Hopital's rule Let and be functions that are differentiable on an open interval containing , except possibly at itself. Assume that for all near except possibly at itself. If lim produces the indeterminate form , then lim lim provided the limit on the right exists or is infinite. This result also applies if lim produces any of the indeterminate forms , or In general, if to to so, near , we have , and the tangent lines to the graphs at have equations: MTH 174, section 4.4, page 1 Example: lim The table suggests that the limit might be , as does the graph: By L'H, we have lim Example: lim by factor and reduce: lim by L'Hopital's Rule: Example:. lim Example:. sin lim sin lim lim lim cos lim lim lim lim sin lim sin lim Since both numerator and denominator are at , we can use L'Hopital's rule: lim lim Now both are still at , so use L'Hopital again: MTH 174, section 4.4, page 2 Example:. lim ln Since both numerator and denominator are at , we can use L'Hopital's rule: lim lim Example: sin sin can use L'Hopital's rule: lim lim cos cos lim cos cos Since both numerator and denominator are at , we Example:. Since both numerator and denominator are at , we can use L'Hopital's rule: lim Example:. lim tan (Remember the degree rule about horizontal asymptotes?) As , tan , so we must rewrite the product as or to . I chose , as follows: a quotient in which lim tan both num. and den. go either to sec lim sec Example:. lim As , the base goes to and the exponent to , so we rewrite usings logs: ln ln lim lim ln lim ln lim ln lim ln . In fact, this limit is the definition of . MTH 174, section 4.4, page 3 Example:. lim has the form . If rewritten as one fraction, we have lim which has the form , so that L'Hopital can be applied. lim Example:. lim sin lim lim From the graph, we see that the limit is . ln Example: ln lim sin lim lim ln sin lim sin consider ln ln ln ln lim ln lim ln lim ln lim lim lim lim MTH 174, section 4.4, page 4 ...
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- Spring '09
- lim, Limit of a function, Indeterminate form, lim lim, lim lim lim