Question

The value of $\displaystyle\lim _{x \rightarrow-\infty}\left\{\frac{x^{4} \sin \left(\frac{1}{x}\right)+x^{2}}{1+\left|x^{3}\right|}\right\}$ is

• A. 1
• B. -1
• C. 0
• D. $\infty$

Answer 1

Answer: (B)

We have, $\displaystyle\lim _{x \rightarrow-\infty}\left\{\frac{x^{4} \sin \left(\frac{1}{x}\right)+x^{2}}{1+\left|x^{3}\right|}\right\}$
$=\displaystyle\lim _{y \rightarrow \infty} \frac{y^{4} \sin \left(\frac{-1}{y}\right)+y^{2}}{1+\left|-y^{3}\right|}$,
where $y=-x$
$=\displaystyle\lim _{y \rightarrow \infty} \frac{-y^{4} \sin \left(\frac{1}{y}\right)+y^{2}}{1+y^{3}}$
$=\displaystyle\lim _{y \rightarrow \infty} \frac{-y \sin \left(\frac{1}{y}\right)+\frac{1}{y}}{\frac{1}{y^{3}}+1}$
$=\frac{-1+0}{0+1}=-1$