The value of $\underset{x→-∞}{\lim}\frac{x^4\sin(\frac{1}{x})+x^2}{1+|x|^3}$ is equal to

-1

$\underset{x→-∞}{\lim}\frac{x^4\sin(\frac{1}{x})+x^2}{1+|x|^3}$

Put $x=-\frac{1}{y}$, ∴ y → 0 and y is positive when x → -∞

$=\underset{y→0}{\lim}\frac{\left(\frac{1}{y^4}\right)\sin(-y)+\left(\frac{1}{y^2}\right)}{1+\left|\left(-\frac{1}{y}\right)\right|}=\underset{y→0}{\lim}\frac{\frac{(y^2-\sin y)}{y^4}}{\left[1+\left(\frac{1}{y^3}\right)\right]}$

$=\underset{y→0}{\lim}\frac{y^2-\sin y}{y^4+y}$   $\left(form\frac{0}{0}\right)$

$=\underset{y→0}{\lim}\frac{2y-\cos y}{4y^3+1}=\frac{0-1}{0+1}=-1$