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2311lectureoutline4.4

2311lectureoutline4.4 - evaluate rather complicated limits...

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MAC2311, Calculus I 4.4 Indeterminate Forms and L’Hopital’s Rule L’Hopital’s Rule: Suppose f and g are differentiable functions and ( ) 0 g x on an open interval I that contains a (except possibly at a). Suppose that 0 ( ) 0 ( ) lim x a g x f x = or ( ) ( ) lim x a g x f x ±∞ ±∞ = (the limit yields an indeterminate form). Then ( ) ( ) ( ) ( ) lim lim x a x a g x g x f x f x = if the limit on the right side exists (or is or -∞ ). L’Hopital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. The rule is valid for one-sided limits and for limits at infinity. The rule allows us to
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Unformatted text preview: evaluate rather complicated limits in a fairly simple manner. It’s a great short cut! While we can safely say that , , k and ⋅∞ = ∞ ∞+∞ =∞ -∞-∞ =-∞ , there are some other indeterminate forms besides or ∞ ∞ . Here are a few: , , 1 , 0 , 0 ∞ ∞ ∞-∞ ⋅∞ . When these indeterminate forms occur, we will attempt to rewrite the expressions as quotients, then evaluate the limit again. If the limit yields or ∞ ∞ , we can apply L’Hopital’s Rule. Try exercises 2, 6, 10, 20, 28, 34,40 on pp. 304-305 in your text....
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