007A Sample Midterm 2, Problem 1 Detailed Solution

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Evaluate the following limits.

(a) Find  

(b) Find  

(c) Evaluate  

Background Information:  
2. Squeeze Theorem
       Let    and    be functions on an open interval    containing   
       such that for all    in  
       If    then  



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


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



Step 1:  
First, recall that
for all  
Then, for all  
Hence, for all  
Step 2:  
Now, notice
Step 3:  
we have
by the Squeeze Theorem.

Final Answer:  

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