MATH104 December 2014
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!
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.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
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 .
Click here for similar questions
MER QBH flag, MER QBS flag, MER QGQ flag, MER RT flag, MER Tag Asymptote