LHopital's Rule

Proof of L'Hopital's Rule

Goal:  Easily find 

       


Suppose that f and g are continuous functions and 

        f(a) = g(a) = 0


Recall the mean value theorem states that

so that

Let 

        a = 0

Then 

       

so that

Hence 


       

L'Hopital's Rule Let            f(c) = g(c) = 0  then

Example:

Exercises:  Determine the following limits if they exist

A)     

B)     

C)     

Hidden Forms of L'Hopital's Rule

We can also use L'Hopital's rule when we have expressions of the form 

Example:  (0)(infinity) 

Example:  (infinity)⁰

Example:  infinity - infinity 

       

which is undefined.

Exercise

Evaluate

       

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