L'Hopital's rule

L'Hôpital's Rule is a method in differential calculus for calculating the limit of a quotient of two functions wherein the entire expression approaches an indeterminate form (e.g. 0/0, infinity/infinity). In the event that this is the case, the limit is equal to the limit of the quotient of the first derivatives of the two functions. Should this also yield an indeterminate form, the process is repeated until a meaningful result is obtained.

Math form

<math>g'(x)\ne0</math> C is some number such that <math>f(c)=0</math> <math>g(c)=0</math> <math>\lim_{x\rightarrow c} \frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}</math>

L'Hopital's Rule is not to be confused with the quotient rule, which allows for the calculation of the derivative of a single function that contains a quotient.

Examples

A standard application of L'Hopital's rule is in evaluating the limit

<math>\lim_{x \to 0} \frac{\sin x}{x}.</math>

In the preceding notation, this is the situation with <math>f(x) = \sin x</math> and <math>g(x) = x</math>. Both the numerator and the denominator tend to 0 as <math>x</math> tends to 0, i.e., <math>\lim_{x \to 0} \sin x = \lim_{x \to 0} x = 0</math>, and so L'Hôpital's rule implies that

<math>

\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\frac{d}{dx} \sin x}{\frac{d}{dx} x} = \lim_{x \to 0} \frac{\cos x}{1} = \lim_{x \to 0} \cos x = \cos 0 = 1. </math>

Outside Links

• The Complete Idiot's Guide to Calculus