l’Hôpital’s rule

L’Hôpital’s rule states that given an unresolvable limit of the form $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$, 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

$$\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\underset{x\to c}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

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

$$\underset{x\to \mathrm{\infty }}{lim}\frac{x^2}{e^x}.$$

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

$$\underset{x\to \mathrm{\infty }}{lim}\frac{x^2}{e^x}=\underset{x\to \mathrm{\infty }}{lim}\frac{2x}{e^x}=\underset{x\to \mathrm{\infty }}{lim}\frac{2}{e^x}=0.$$

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