007A Sample Midterm 1, Problem 2 Detailed Solution

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Consider the following function  

(a) Find  

(b) Find  

(c) Find  

(d) Is    continuous at    Briefly explain.


Foundations:  
1. If  
        then  
2.    is continuous at    if
       


Solution:

(a)

Step 1:  
Notice that we are calculating a left hand limit.
Thus, we are looking at values of    that are smaller than  
Using the definition of    we have
       
Step 2:  
Now, we have

       

(b)

Step 1:  
Notice that we are calculating a right hand limit.
Thus, we are looking at values of    that are bigger than  
Using the definition of    we have
       
Step 2:  
Now, we have

       

(c)

Step 1:  
From (a) and (b), we have
       
and
       
Step 2:  
Since
       
we have
       

(d)

Step 1:  
From (c), we have
       
Also,
       
Step 2:  
Since
       
 is continuous at  


Final Answer:  
    (a)    
    (b)    
    (c)    
    (d)       is continuous at    since  

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