009A Sample Final 1, Problem 1

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In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

(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 factoring the numerator. We have

       

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

       

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

       

(b)

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

       

Step 2:  
This limit is  

(c)

Step 1:  
We have

       

Since we are looking at the limit as    goes to negative infinity, we have  
So, we have

       

Step 2:  
We simplify to get

       


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

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