# L'Hopital's rule

This is the current revision of L'Hopital's rule as edited by TK (Talk | contribs) at 00:32, 26 January 2010. This URL is a permanent link to this version of this page.

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 of 0/0 or 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 (provided that limit exists). Should this also yield an indeterminate form, the process is repeated until a meaningful result is obtained.[1]

  C is some number such that   

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

### Example 1

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



In the preceding notation, this is the situation with  and . Both the numerator and the denominator tend to 0 as  tends to 0, i.e., , and so L'Hôpital's rule implies that



### 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 asymptotes of rational functions. For example, suppose we seek to compute



This is an indeterminate form . Applying L'Hopital's rule once yields



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



We conclude that



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



where  and  are the leading coefficients of  and  (i.e., the coefficients on the term  in these two polynomials). The example given is a case of this fact with  (since both  and  are quadratic), and with  and .

### Example 3

We can use L'Hôpital's rule to prove the following:



This is another example where the limit is in the form of .
Proof

For Integer n:

For non-integer n:

(where  is the floor function of n and  is the ceiling function of n)
Since ,  and therefore

This completes the proof.