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'''Discovered By''' | |||
Guillaume de l'Hopital | Guillaume de l'Hopital | ||
'''The Rule''' | |||
For <math> \frac {0}{0} </math> or <math> \frac {\infty}{\infty} </math> case, the limit <cmath> \lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)} </cmath> where <math> f'(x) </math> and <math> g'(x) </math> are the first derivatives of <math> f(x) </math> and <math> g(x) </math>, respectively. | For <math> \frac {0}{0} </math> or <math> \frac {\infty}{\infty} </math> case, the limit <cmath> \lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)} </cmath> where <math> f'(x) </math> and <math> g'(x) </math> are the first derivatives of <math> f(x) </math> and <math> g(x) </math>, respectively. | ||
'''Examples''' | |||
<math> \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} </math> | <math> \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} </math> | ||
<math> \lim_{x \to 0} \cfrac {\sin x}{x} = \lim_{x \to 0} \cfrac {\cos x}{1} = \cos (0) = \boxed {1} </math> | <math> \lim_{x \to 0} \cfrac {\sin x}{x} = \lim_{x \to 0} \cfrac {\cos x}{1} = \cos (0) = \boxed {1} </math> | ||
Revision as of 01:33, 2 March 2011
Discovered By
Guillaume de l'Hopital
The Rule
For 
or 
case, the limit ![\[\lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)}\]](http://latex.artofproblemsolving.com/7/c/f/7cfddacb9978addb09ea544e0b4f22c716eb236c.png)
where 
and 
are the first derivatives of 
and 
, respectively.
Examples
