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

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: $$\therefore\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}=\infty$$

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