Science:Math Exam Resources/Courses/MATH110/December 2011/Question 01 (a)

MATH110 December 2011

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Question 01 (a)
Evaluate the following limit. $\lim _{x\to -\infty }{\frac {5x^{3}-2x^{2}+1}{6x^{3}+7}}$

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What can you do when dealing with a limit at infinity of a rational function?

Hint 2
Only the leading term of a polynomial matters when considering what happens at infinity.

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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. When considering limits at infinity of rational functions, only the dominant term matters: ${\begin{aligned}\lim _{x\to -\infty }{\frac {5x^{3}-2x^{2}+1}{6x^{3}+7}}&=\lim _{x\to -\infty }{\frac {x^{3}\left(5-{\frac {2}{x}}+{\frac {1}{x^{3}}}\right)}{x^{3}\left(6+{\frac {7}{x^{3}}}\right)}}\\&=\lim _{x\to -\infty }{\frac {5-{\frac {2}{x}}+{\frac {1}{x^{3}}}}{6+{\frac {7}{x^{3}}}}}\\&={\frac {5}{6}}\end{aligned}}$ Here we factored out the leading term (the $x^{3}$) on both the top and bottom. That allowed us to cancel it and then evaluate the rest of the limit. The $1/x^{n}$ terms go to zero as $x\to \infty$.

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