009A Sample Final 3, Problem 7

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Compute

(a)  

(b)  

(c)  

Foundations:  
L'Hôpital's Rule
        Suppose that    and    are both zero or both  

        If    is finite or  

        then  


Solution:

(a)

Step 1:  
We begin by noticing that we plug in    into
       
we get  
Step 2:  
Now, we multiply the numerator and denominator by the conjugate of the denominator.
Hence, we have
       

(b)

Step 1:  
We proceed using L'Hôpital's Rule. So, we have

       

Step 2:  
Now, we plug in    to get
       

(c)

Step 1:  
We begin by factoring the numerator and denominator. We have

       

So, we can cancel    in the numerator and denominator. Thus, we have

       

Step 2:  
Now, we can just plug in    to get
       


Final Answer:  
   (a)   
   (b)   
   (c)   

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