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Question 05 (e) 

Let . Find the horizontal asymptote and the vertical asymptote of . Note that . 
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Hint 

How are limits and horizontal asymptotes related? 
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Solution 

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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 . 