Section4-5 - Section 4.5 Indeterminate Forms and...

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1 Section 4.5 Indeterminate Forms and l’Hospital’s Rule • Goals – Introduce the various types of indeterminate forms – Find limits of indeterminate forms using l’Hospital’s Rule . Introduction • Suppose we want to analyze the behavior of the function •S ince… – the limit of the denominator as x 1 is 0 , we cannot use Limit Laws to find – the limit of the numerator is also 0 as x 1 , it is not even clear whether the limit exists at all. () == ln near 1. 1 x Fx x x ( ) 1 lim ; x L’Hospital’s Rule • Here is a way to evaluate such limits: L’Hospital’s Rule (cont’d) • Some remarks: – L’Hospital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives . – Be sure to verify the required conditions on f and g before using l’Hospital’s Rule. – L’Hospital’s Rule is also valid for one-sided limits and for limits at or – .
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2 Example • Find the limit • Solution Since we can apply l’Hospital’s Rule: 1 ln lim mentioned earlier. 1 x x x ( ) →→ == = 11 limln ln1 0 and lim 1 0 xx Example •Ca lcu la te • Solution We have so l’Hospital’s Rule gives •S ince the limit on the right is still indeterminate. →∞ 2 lim . x x e x →∞ →∞ =∞ 2 lim and lim , x ex →∞ →∞ = 2 lim lim 2 ee →∞ and 2 as , x x Solution (cont’d) • However, a second application of
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This note was uploaded on 04/18/2008 for the course MATH 210 taught by Professor Zhoramanseur during the Spring '08 term at SUNY Oswego.

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Section4-5 - Section 4.5 Indeterminate Forms and...

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