Answer
$$\lim_{x\to\infty}\frac{x^{-1}+x^{-4}}{x^{-2}-x^{-3}}=\infty$$
Work Step by Step
We would treat these exercises just like we treat the limits of rational functions: we would divide both numerator and denominator by the highest power of $x$ in the denominator.
Also remember that $$\lim_{x\to\infty}\frac{a}{x^n}=\lim_{x\to-\infty}\frac{a}{x^n}=a\times0=0\hspace{1cm}(a\in R)$$
$$A=\lim_{x\to\infty}\frac{x^{-1}+x^{-4}}{x^{-2}-x^{-3}}$$
The highest power of $x$ in the denominator here is $x^{-2}$, so we divide both numerator and denominator by $x^{-2}$: $$A=\lim_{x\to\infty}\frac{\frac{x^{-1}}{x^{-2}}+\frac{x^{-4}}{x^{-2}}}{1-\frac{x^{-3}}{x^{-2}}}=\lim_{x\to\infty}\frac{x^{-1+2}+x^{-4+2}}{1-x^{-3+2}}=\lim_{x\to\infty}\frac{x+x^{-2}}{1-x^{-1}}$$
$$A=\lim_{x\to\infty}\frac{x+\frac{1}{x^2}}{1-\frac{1}{x}}$$
$$A=\frac{\lim_{x\to\infty}x+0}{1-0}=\frac{\lim_{x\to\infty}x}{1}=\lim_{x\to\infty}x=\infty$$
