We want to prove $\lim_{x\to +\infty}\frac{x^3}{|x|} = \infty$. I don't think the usual "$\varepsilon-\delta$ definition would work since we are dealing with $\infty$. How else could I approach this problem?
I know plugging in $\infty$ would show this. But is there a more rigorous proof?
For example, if I wanted to change this problem so that $x\to0$ the plugging-in method would not work anymore since we cannot divide by $0$. So there must be a way to solve this so that this problem does not arise!
Thank you.