calculus - Prove $lim_{xtoinfty} frac{x}{(ln x)^3} = infty$

$$\lim_{x \to \infty} \frac{x}{(\ln x)^3} = \infty$$

One way to think of this problem is in terms of the relative growth rates between the numerator and denominator. I know that $x$ grows asymptotically faster than $(\ln x)^3$ according to WolframAlpha. How can I prove this?

Answer

$ \lim_{x \to \infty} \frac{x}{(\ln x)^3}=\quad\quad (\frac{\infty}{\infty}\textrm {form, using L'Hospital rule})\\ =\lim_{x \to \infty}\frac{x}{3(\ln x)^2}\quad\quad (\frac{\infty}{\infty}\textrm {form, using L'Hospital rule})\\=\lim_{x \to \infty}\frac{x}{6(\ln x)}\quad\quad (\frac{\infty}{\infty}\textrm {form, using L'Hospital rule})\\ =\lim_{x \to \infty}\frac{x}{6}=\infty$