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25-LHopital

25-LHopital - Math 1a L’Hˆopital’s Rule Fall 2009...

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Unformatted text preview: Math 1a L’Hˆopital’s Rule Fall, 2009 ’ & $ % Indeterminate Forms We’re considering lim x → a f ( x ) g ( x ) . We begin with several indeterminate forms: Type : lim x → a f ( x ) = 0 and lim x → a g ( x ) = 0 Type ∞ ∞ : lim x → a f ( x ) = ∞ or- ∞ and lim x → a g ( x ) = ∞ or- ∞ L’Hˆ opital’s Rule (or L’Hospital’s Rule ) is: if f and g are differentiable and g ( x ) is not zero near a (but g ( a ) may be zero). Then if the limit (on the left, below) is of indeterminate form 0 / 0 or ∞ / ∞ , then lim x → a f ( x ) g ( x ) = lim x → a f ( x ) g ( x ) provided the limit on the right exists (or is + ∞ or-∞ ). 1 For each of the following, verify that the given limit is in an indeterminate form, then use L’Hˆ opital’s Rule (if possible) to determine the limit. (a) lim x → sin( x ) x (b) lim x → sin( x ) x 2 (c) lim x → + sin( x ) √ x (d) lim x → 1 x 2- 1 ln( x ) (e) lim x → e x- 1 x (f) lim x →∞ ln( x ) √ x (g) lim x →...
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25-LHopital - Math 1a L’Hˆopital’s Rule Fall 2009...

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