Science:Math Exam Resources/Courses/MATH104/December 2012/Question 02 (d)
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Question 02 (d) 

Consider the function Its first and second derivatives are given by Write the equation of all vertical asymptotes (if there are any). Show that is a slant asymptote. 
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 

At what values of does have possible vertical asymptotes? If you think that has a vertical asymptote at , then what does satisfy? If is a slant asymptote, then what does the difference between and become as gets large? Show this explicity. 
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Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We know that does not exist where its denominator equals 0: So there are possible vertical asymptotes at . To confirm that there is a vertical asymptote at , we must show that This is easy to show because as from the left (from values of less than ), the numerator of approaches and the denominator approaches 0 from the left (negative values). So we know that Similarly we can show that the following hold: Therefore, there are two vertical asymptotes:
Thus . Therefore, y = (2/3)x is a slant asymptote of f(x). 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Proceed as in solution 1 to find the vertical asymptotes. If is a slant (oblique) asymptote to then the difference between and the slant asymptote has to vanish when x gets very small or very large: We will show that the first equation is true. Similarly, it can be shown that Therefore, y = (2/3)x is a slant asymptote of f(x). 