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| − | Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval <i>I</i> containing a. | + | Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval <i>I</i> containing a. <br> |
| + | Suppose also that <math>g'(x)\neq0</math> on <i>I</i> if <math>x\neq a</math>. <br> | ||
| + | Then <br> | ||
<math> | <math> | ||
\lim_{x \to\ a}\frac{f(x)}{g(x)}= \lim_{x \to\ a}\frac{f'(x)}{g'(x)} | \lim_{x \to\ a}\frac{f(x)}{g(x)}= \lim_{x \to\ a}\frac{f'(x)}{g'(x)} | ||
</math>, <br> | </math>, <br> | ||
| − | if the | + | if the limit on the right exists (or is <math>\infty</math> or -<math>\infty</math> |
). | ). | ||
This is Elizabeth's favorite theorem. | This is Elizabeth's favorite theorem. | ||
Revision as of 11:49, 4 September 2008
Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval I containing a.
Suppose also that $ g'(x)\neq0 $ on I if $ x\neq a $.
Then
$ \lim_{x \to\ a}\frac{f(x)}{g(x)}= \lim_{x \to\ a}\frac{f'(x)}{g'(x)} $,
if the limit on the right exists (or is $ \infty $ or -$ \infty $
).
This is Elizabeth's favorite theorem.
