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

We have,

$\begin{array}{l}\underset{x\to -\mathrm{\infty }}{lim} \left\{\frac{x^4\sin \left(\begin{array}{c}1\\ x\end{array}\right)+x^2}{1+|x{|}^3}\right\}\\ =\underset{y\to \mathrm{\infty }}{lim} \frac{y^4\sin \left(\frac{-1}{y}\right)+y^2}{1+\left|-y^3\right|},\text{ where }y=-x\\ =\underset{y\to \mathrm{\infty }}{lim} \frac{-y^4\sin \left(\frac{1}{y}\right)+y^2}{1+y^3}\\ =\underset{y\to \mathrm{\infty }}{lim} \frac{-y\sin \left(\frac{1}{y}\right)+\frac{1}{y}}{\frac{1}{3}+1}=\frac{-1+0}{0+1}=-1\end{array}$