Difference between revisions of "007A Sample Midterm 1, Problem 1 Detailed Solution"

From Math Wiki
Jump to navigation Jump to search
 
Line 8: Line 8:
 
<hr>
 
<hr>
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations: &nbsp;  
+
!Background Information: &nbsp;  
 
|-
 
|-
 
| '''1.''' If &nbsp;<math style="vertical-align: -12px">\lim_{x\rightarrow a} g(x)\neq 0,</math>&nbsp; we have
 
| '''1.''' If &nbsp;<math style="vertical-align: -12px">\lim_{x\rightarrow a} g(x)\neq 0,</math>&nbsp; we have

Latest revision as of 12:27, 5 January 2018

Find the following limits:

(a) Find    provided that  

(b) Find  

(c) Evaluate  


Background Information:  
1. If    we have
       
2. Recall
       


Solution:

(a)

Step 1:  
Since  
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

       

(b)

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

(c)

Step 1:  
When we plug in values close to    into  
we get a small denominator, which results in a large number.
Thus,
       
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:  
    (a)    
    (b)    
    (c)    

Return to Sample Exam