Calculus/L'Hôpital's Rule

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L'Hôpital's Rule


L'Hôpital's RuleEdit

Occasionally, 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.

Definition: Indeterminate Limit
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.

TheoremEdit

If   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   caseEdit

Suppose 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 3.8) 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  .

ExamplesEdit

Example 1Edit

Find  

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.

Example 2Edit

Find  

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 3Edit

Find  

This time, plugging in   for x gives you   . So using L'Hôpital's rule gives:

 

Therefore,   is the answer.

Example 4Edit

Find  

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

 

NoteEdit

This does not prove that   because using the same method,

 

ExercisesEdit

Evaluate the following limits using L'Hôpital's rule:

1.  
 
 
2.  
 
 
3.  
 
 
4.  
 
 
5.  
 
 

Solutions

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L'Hôpital's Rule