Evaluate: $\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$ Find the following limit $$\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$$
How do I find this limit? If I had to guess I'd say it converges to $-2$ but the usual things like L'Hôpital or clever factorisation don't seem to work in this case.

$\displaystyle \lim_{x \to \infty} \,\, (x^3-1)^{\frac{1}{3}} - (x^{3})^{\frac{1}{3}} - 2=\lim_{x \to \infty}\frac{x^3-1-x^3}{(x^3-1)^{\frac{2}{3}}+(x^3-1)^{\frac{1}{3}}x^{\frac{1}{3}}+x^{\frac{2}{3}}}-2=-2$
