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Section7_7_review - Section 7.7 Indeterminate Forms and...

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Section 7.7: Indeterminate Forms and L’Hopital’s Rule L’Hopital’s Rule: Let f and g be differentiable functions with g x (29 0 near a (except possibly at a ). If lim x a f x ( 29 g x ( 29 is an indeterminate quotient, then lim x a f x (29 gx (29 H lim x a f x (29 g x (29 where an indeterminate quotient is of the form 0 0 or ± ∞ ± ∞ . Comments: i) lim x a in L’Hopital’s Rule can be replaced with lim x a + , lim x a - , lim x →∞ , and lim x →-∞ . In other words, L’Hopital’s Rule is valid for one-sided limits as well as x to a constant, to infinity, and to negative infinity. ii) L’Hopital’s Rule can be used several times in succession, a long as you have an indeterminate quotient each time. iii) L’Hopital’s Rule is only valid for indeterminate quotients. If you have an indeterminate product (such as 0 ⋅∞ or 0 ⋅ -∞ ( 29 ), an indeterminate difference (such as ∞-∞ ), or an indeterminate power (such as 1 , 0 , or 0 0 ), the expression must first be converted to a quotient before L’Hopital’s Rule can be used.
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