# l’Hôpital’s rule

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 $c$ as the ratio $\frac{f^{\prime}(x)}{g^{\prime}(x)}$. In short, if the limit of a ratio of functions approaches an indeterminate form, then

 $\lim_{x\rightarrow c}\frac{f(x)}{g(x)}=\lim_{x\rightarrow c}\frac{f^{\prime}(x% )}{g^{\prime}(x)}$

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 $\lim\frac{f^{\prime}(x)}{g^{\prime}(x)}$ does not prove the nonexistance of $\lim\frac{f(x)}{g(x)}$.

Example: We try to determine the value of

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

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

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

Another example of the usage of L’Hôpital’s rule can be found http://planetmath.org/node/5741here.

 Title l’Hôpital’s rule Canonical name LHopitalsRule Date of creation 2013-03-22 12:28:15 Last modified on 2013-03-22 12:28:15 Owner mathwizard (128) Last modified by mathwizard (128) Numerical id 13 Author mathwizard (128) Entry type Theorem Classification msc 26A24 Classification msc 26C15 Synonym l’Hospital’s rule Related topic IndeterminateForm Related topic DerivationOfHarmonicMeanAsTheLimitOfThePowerMean Related topic ImproperLimits Related topic ExampleUsingStolzCesaroTheorem