Calculus/L'Hôpital's Rule
L'Hôpital's Rule
editOccasionally, one comes across a limit which results in or , which are called indeterminate limits. However, it is still possible to solve these by using L'Hôpital's rule. This rule is vital in explaining how other limits can be derived.
If exists, where or , the limit is said to be indeterminate.
All of the following expressions are indeterminate forms.
These expressions are called indeterminate because you cannot determine their exact value in the indeterminate form. Depending on the situation, each indeterminate form could evaluate to a variety of values.
Theorem
editIf is indeterminate of type or ,
then , where .
In other words, if the limit of the function is indeterminate, the limit equals the derivative of the top over the derivative of the bottom. If that is indeterminate, L'Hôpital's rule can be used again until the limit isn't or .
Proof of the 0/0 case
editSuppose that for real functions and , and that exists. Thus and exist in an interval around , but maybe not at itself. Thus, for any , in any interval or , and are continuous and differentiable, with the possible exception of . Define
Note that , , and that are continuous in any interval or and differentiable in any interval or when .
Cauchy's Mean Value Theorem (see 3.9) tells us that for some or . Since , we have for .
Since or , by the squeeze theorem
This implies
So taking the limit as of the last equation gives , which is equivalent to the more commonly used form .
Examples
editExample 1
editFind
Since plugging in 0 for x results in , use L'Hôpital's rule to take the derivative of the top and bottom, giving:
Plugging in 0 for x gives 1 here. Note that it is logically incorrect to prove this limit by using L'Hôpital's rule, as the same limit is required to prove that the derivative of the sine function exists: it would be a form of begging the question, or circular reasoning. An alternative way to prove this limit equal one is using squeeze theorem.
Example 2
editFind
First, you need to rewrite the function into an indeterminate limit fraction:
Now it's indeterminate. Take the derivative of the top and bottom:
Plugging in 0 for once again gives 1.
Example 3
editFind
This time, plugging in for x gives you . So using L'Hôpital's rule gives:
Therefore, is the answer.
Example 4
editFind
Plugging the value of x into the limit yields
- (indeterminate form).
Let
We now apply L'Hôpital's rule by taking the derivative of the top and bottom with respect to .
Since
We apply L'Hôpital's rule once again
Therefore
And
Similarly, this limit also yields the same result
Note
editThis does not prove that because using the same method,
Exercises
editEvaluate the following limits using L'Hôpital's rule: