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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.

L'Hôpital'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 ==

== Examples ==

===Example 1===

===Example 1===

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

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

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

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

===Example 2===

===Example 2===

L'Hopital's rule may also be used in the evaluation of the indeterminate form infinity/infinity.  This version of the rule is useful in computing the horizontal [[asymptote|asymptotes]] of rational functions.  For example, suppose we seek to compute

L'H&ocirc;pital's rule may also be used in the evaluation of the indeterminate form infinity/infinity.  This version of the rule is useful in computing the horizontal [[asymptote|asymptotes]] of rational functions.  For example, suppose we seek to compute

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7}.</math>

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7}.</math>

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = \lim_{x \to \infty} \frac{\frac{d}{dx}(2x^2+3x+2)}{\frac{d}{dx}(x^2-5x+7)} = \lim_{x \to \infty} \frac{4x+3}{2x-5}.</math>

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = \lim_{x \to \infty} \frac{\frac{d}{dx}(2x^2+3x+2)}{\frac{d}{dx}(x^2-5x+7)} = \lim_{x \to \infty} \frac{4x+3}{2x-5}.</math>

This is still an indeterminate form.  To evaluate the limit, it is necessary to invoke L'Hopital's rule a second time:

This is still an indeterminate form.  To evaluate the limit, it is necessary to invoke L'H&ocirc;pital's rule a second time:

::<math>\lim_{x \to \infty} \frac{4x+3}{2x-5} = \lim_{x \to \infty} \frac{\frac{d}{dx}(4x+3)}{\frac{d}{dx}(2x-5)} = \lim_{x \to \infty} \frac{4}{2} = 2.</math>

::<math>\lim_{x \to \infty} \frac{4x+3}{2x-5} = \lim_{x \to \infty} \frac{\frac{d}{dx}(4x+3)}{\frac{d}{dx}(2x-5)} = \lim_{x \to \infty} \frac{4}{2} = 2.</math>

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = 2.</math>

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = 2.</math>

An easy extension of this argument is useful for finding horizontal asymptotes of more general rational functions.  Suppose that <math>f</math> and <math>g</math> are two polynomials of equal degree <math>n</math>.  Applying L'Hopital's rule <math>n</math> times we may discover that

An easy extension of this argument is useful for finding horizontal asymptotes of more general rational functions.  Suppose that <math>f</math> and <math>g</math> are two polynomials of equal degree <math>n</math>.  Applying L'H&ocirc;pital's rule <math>n</math> times we may discover that

::<math>\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{f_n}{g_n},</math>

::<math>\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{f_n}{g_n},</math>

== Outside Links ==

== Outside Links ==

*[http://books.google.com/books?id=H-0L9Dxor6sC&pg=PA144&lpg=PA144&dq=L%27H%C3%B4pital%27s+rule&source=bll&ots=3HfHZW3-DZ&sig=Ripjx5A7yw-OcrHN91sxEN8Gr5U&hl=en&ei=kg2WSePdKYKUsQOT_JiuBw&sa=X&oi=book_result&resnum=11&ct=result The Complete Idiot's Guide to Calculus]

*[http://books.google.com/books?id=H-0L9Dxor6sC&pg=PA144&lpg=PA144&dq=L%27H%C3%B4pital%27s+rule&source=bll&ots=3HfHZW3-DZ&sig=Ripjx5A7yw-OcrHN91sxEN8Gr5U&hl=en&ei=kg2WSePdKYKUsQOT_JiuBw&sa=X&oi=book_result&resnum=11&ct=result The Complete Idiot's Guide to Calculus]

== References ==

== References ==

{{reflist}}

{{reflist}}