PlanetMath: l'Hôpital's rule

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l'Hôpital's rule

(Theorem)

L'Hôpital's rule states that given an unresolvable limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ , the ratio of functions $\frac{f(x)}{g(x)}$ will have the same limit at as the ratio $\frac{f'(x)}{g'(x)}$ . In short, if the limit of a ratio of functions approaches an indeterminate form, then

$\displaystyle \lim_{x\rightarrow c}\frac{f(x)}{g(x)} = \lim_{x\rightarrow c}\frac{f'(x)}{g'(x)}$

provided this last limit exists. L'Hôpital's rule may be applied indefinitely as long as the conditions are satisfied. However it is important to note, that the nonexistance of $\lim\frac{f'(x)}{g'(x)}$ does not prove the nonexistance of $\lim\frac{f(x)}{g(x)}$ .

Example: We try to determine the value of

$\displaystyle \lim_{x\to \infty}\frac{x^2}{e^x}.$

approaches $\infty$ the expression becomes an indeterminate form $\frac{\infty}{\infty}$ . By applying L'Hôpital's rule twice we get

$\displaystyle \lim_{x\to\infty}\frac{x^2}{e^x}=\lim_{x\to \infty}\frac{2x}{e^x}=\lim_{x\to \infty}\frac{2}{e^x}=0.$

Another example of the usage of L'Hôpital's rule can be found here.

"l'Hôpital's rule" is owned by mathwizard. [ full author list (2) | owner history (1) ]

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Other names:

l'Hospital's rule

Attachments:: proof of De l'Hôpital's rule (Proof) by paolini
proof of l'Hôpital's rule for $\infty/\infty$ form (Proof) by stevecheng

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Cross-references: expression, indeterminate form, functions, ratio, limit
There are 9 references to this entry.

This is version 10 of l'Hôpital's rule, born on 2002-02-25, modified 2004-04-07.
Object id is 2657, canonical name is LHpitalsRule.
Accessed 24515 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26C15 (Real functions :: Polynomials, rational functions :: Rational functions)

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