009A Sample Final 3, Problem 1

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Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

(a)  

(b)    given that  

(c)  


Foundations:  
1. If    we have
       
2.  


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:  
Since  
we have
       
Step 2:  
If we multiply both sides of the last equation by    we get
       
Now, using properties of limits, we have
       
Step 3:  
Solving for    in the last equation,
we get

       

(c)

Step 1:  
First, we write
       
Step 2:  
Now, we have
       


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

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