Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 03 (a)
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Question 03 (a) 

(a) Let . Find and justify the vertical asymptotes of . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The vertical asymptotes are the vertical lines , where either , , or . 
Hint 2 

The candidates of are the roots of the quadratic equation . Note that the left hand side can be factorized into two linear factors. 
Hint 3 

The numerator can also be factorized. Pay attention to the factor that can be cancelled. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let us first simplify the function: Note that is defined for all real numbers except at (where ) and (where ).
The candidates for vertical asymptotes are and . However we need to compute the limits of at these values to check whether each is a vertical asymptote. For , we can write From this we see that
and Therefore, is a vertical asymptote, while is not one. Answer: The (only) vertical asymptote is . 