Science:Math Exam Resources/Courses/MATH104/December 2014/Question 05 (e)
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 • Q3 (a) • Q3 (b) • Q3 (c) • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q5 (e) • Q5 (f) • Q6 •
Question 05 (e) 

Let . Find the horizontal asymptote and the vertical asymptote of . Note that . 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

How are limits and horizontal asymptotes related? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. We have already determined what the vertical asymptote is in part , by seeing where is undefined. To see the behaviour of as it approaches from the right and left, we have the following calculations:
where we obtain the last equality since the square of any negative number is a positive number. This happens when so that is our vertical asymptote. To find the horizontal asymptote, we find and . Essentially, what we are determining is the behaviour of as gets very large and very small. We will apply L’Hopital’s rule when finding the two limits.
Since , we have that (the limit does not exist). So as gets very large, also gets very large and proceeds towards .
In the second limit calculation, we applied L’Hopital’s rule twice. We learn from the second calculation that as gets very small and speeds towards , approaches . 