Science:Math Exam Resources/Courses/MATH221/December 2007/Question Section 103 10 (c)

MATH221 December 2007

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Question Section 103 10 (c)
Find the limit $\lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}.$

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

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From part (a), we found that ${\begin{aligned}x_{n}&=2(4)^{n+1}+3(-1)^{n+1}\\y_{n}&=2[4^{n+1}+(-1)^{n}]\end{aligned}}$. Therefore, ${\frac {x_{n}}{y_{n}}}={\frac {2(4)^{n+1}+3(-1)^{n+1}}{2[4^{n+1}+(-1)^{n}]}}={\frac {1+{\frac {3}{2}}(-1/4)^{n+1}}{1+{\frac {1}{4}}(-1/4)^{n}}},$ where we have obtained the last equality by dividing top and bottom by $2(4)^{n+1}$. As $\lim _{n\to \infty }\left({\frac {-1}{4}}\right)^{n}=0,$ we conclude $\lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}=1.$