The limit of $(x^3+\cos x+e^{-2x})/(x^2 \sqrt{x^2+1})$ as $x\to\infty$

Question

I have this infinity problem which I do not know the answer to:

$$\lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2 \sqrt{x^2+1}}$$

I thaught that because $x^3$ is the fastest growing part, this would be infinity, but WolframAlpha says that this will be equal to $1.$

Answer 1

Divide the whole thing by $x^3$ and you get $$\frac{1+\cos x/x^3+e^{-2x}/x^3}{\sqrt{1+1/x^2}}$$

now as $x\to\infty$, all the terms go to $0$, leaving the limit as $1$.

Answer 2

$$\lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}}=\lim_{x\to\infty}\frac{1+\frac{\cos x}{x^3}+\frac{e^{-2x}}{x^3}}{\sqrt{1+\frac{1}{x^2}}}=1$$