009A Sample Midterm 1, Problem 1 Detailed Solution

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Find the following limits:

(a) Find    provided that  

(b) Find  

(c) Evaluate  

Background Information:  
1. If    we have
2. Recall



Step 1:  
we have
Step 2:  
If we multiply both sides of the last equation by    we get
Now, using linearity properties of limits, we have
Step 3:  
Solving for    in the last equation,
we get



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


Step 1:  
When we plug in values close to    into  
we get a small denominator, which results in a large number.
is either equal to    or  
Step 2:  
To figure out which one, we factor the denominator to get
We are taking a right hand limit. So, we are looking at values of  
a little bigger than    (You can imagine values like   )
For these values, the numerator will be negative.
Also, for these values,    will be negative and    will be positive.
Therefore, the denominator will be negative.
Since both the numerator and denominator will be negative (have the same sign),

Final Answer:  

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