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L'Hpital's Rule

Revision as of 01:31, 2 March 2011 by Ahaanomegas (talk | contribs) (l'Hopital's Rule article)
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[b][size=150]Discovered by[/size][/b]

Guillaume de l'Hopital

[b][size=150]The Rule[/size][/b]

For $\frac {0}{0}$ or $\frac {\infty}{\infty}$ case, the limit \[\lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)}\] where $f'(x)$ and $g'(x)$ are the first derivatives of $f(x)$ and $g(x)$, respectively.

[b][size=150]Examples[/size][/b]

$\lim_{x \to 4} \cfrac {x^3 - 64}{4 - x} = \lim_{x \to 4} \cfrac {3x^2}{-1} = -3(4)^2 = -3(16) = \boxed {-48}$

$\lim_{x \to 0} \cfrac {\sin x}{x} = \lim_{x \to 0} \cfrac {\cos x}{1} = \cos (0) = \boxed {1}$

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