Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate. lim_{xrightarrowinfty}frac{4x^3-2x-1}{x^2-1}

Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate. lim_{xrightarrowinfty}frac{4x^3-2x-1}{x^2-1}

Question
Limits and continuity
asked 2021-03-09
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
\(\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}\)

Answers (1)

2021-03-10
We need to compute the value of given limit
\(\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}\)
Dividing both numerator and denominator by \(x^2\) , we get
\(\lim_{x\rightarrow\infty}\frac{\frac{4x^3-2x-1}{x^2}}{\frac{x^2-1}{x^2}}\)
\(=\lim_{x\rightarrow\infty}\frac{4x-\frac{2}{x}-\frac{1}{x^2}}{1-\frac{1}{x^2}}\)
\(=\frac{\lim_{x\rightarrow\infty}(4x-\frac{2}{x}-\frac{1}{x^2})}{\lim_{x\rightarrow\infty}(1-\frac{1}{x^2})}\)
\(=\frac{(\infty-0-0)}{(1-0)}\)
\(=\infty\)
Answer: \(\therefore\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}=\infty\)
0

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