Science:Math Exam Resources/Courses/MATH102/December 2012/Question A 03

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MATH102 December 2012

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Question A 03
Which of the following statements concerning the limit below is necessarily true? $L=\lim _{x\rightarrow -\infty }{\frac {x^{n}-2x^{3}+2}{x^{3}+4}}$ (a) If n < 3 then L = 0. (b) If n = 3 then L = -2. (c) If n > 3 and n is odd then L = $-\infty$ (d) If n > 3 and n is even then L = $-\infty$.

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
Assume the "if" statement of each answer choice is true. Then evaluate the limit to see if it matches the proposed conclusion.

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.

Solution

Let's test each answer one-by-one: (a) If n < 3 then L = 0. If n < 3, try a test value of n = 2. Then our limit becomes: ${\begin{aligned}L&=\lim _{x\rightarrow -\infty }{\frac {x^{2}-2x^{3}+2}{x^{3}+4}}\\&=\lim _{x\rightarrow -\infty }{\frac {-2+x^{2}/x^{3}+2/x^{3}}{1+4/x^{3}}}\\&=-2\end{aligned}}$ The answer does not match the conclusion L = 0 so this answer is not necessarily true. (b) If n = 3 then L = -2. If n = 3 then our limit becomes: ${\begin{aligned}L&=\lim _{x\rightarrow -\infty }{\frac {x^{3}-2x^{3}+2}{x^{3}+4}}\\&=\lim _{x\rightarrow -\infty }{\frac {-1+2/x^{3}}{1+4/x^{3}}}\\&=-1\end{aligned}}$ The answer does not match the conclusion L = -2 so this answer is not true. (c) If n > 3 and n is odd then L = $-\infty$ If n > 3 and n is odd try a test value of n = 5. Then our limit becomes: ${\begin{aligned}L&=\lim _{x\rightarrow -\infty }{\frac {x^{5}-2x^{3}+2}{x^{3}+4}}\\&=\lim _{x\rightarrow -\infty }{\frac {x^{2}-2+2/x^{3}}{1+4/x^{3}}}\\&=\infty \end{aligned}}$ The answer does not match the conclusion $L=-\infty$, so this answer is not necessarily true. (d) If n > 3 and n is even then L = $-\infty$. If n > 3, and even try a test value of n = 4. Then our limit becomes: ${\begin{aligned}L&=\lim _{x\rightarrow -\infty }{\frac {x^{4}-2x^{3}+2}{x^{3}+4}}\\&=\lim _{x\rightarrow -\infty }{\frac {x-2+2/x^{3}}{1+4/x^{3}}}\\&=-\infty \end{aligned}}$ The answer matches the conclusion $L=-\infty$ so this answer is correct. The final answer is (d).

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