MATH104 December 2013
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Question 02 (f)
Consider the function
Its first and second derivatives are given by
(f) Find any horizontal and vertical asymptotes of the function and write their equations.
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!
The function has a vertical asymptote at if
The function has a horizontal asymptote at if
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Points where the numerator goes to infinity or the denominator goes to zero are candidates for where a fraction might have a vertical asymptote. In our case the denominator goes to zero for . To evaluate the corresponding limits, plug in a value that is close to the limiting value on either side:
The sign is positive since the numerator approaches 4 and plugging in a number slightly smaller than -2 yields a very small, positive denominator. Similar considerations yield
Thus there are vertical asymptotes at .
For the horizontal asymptotes, we check the limit of the function as goes to positive and negative infinity. We can do this by factoring out the largest power of the numerator and the denominator.
This tells us that we have horizontal asymptotes in both directions, at .
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