Limit $\lim_{x \rightarrow -\infty} (1 + x^3) = -\infty.$ by epsilon delta

Question

Prove the following using the definition of a limit: $$\lim_{x \rightarrow -\infty} (1 + x^3) = -\infty.$$ I know we have to show that for all $x < N$, there must be $f(x) < M$, but I'm not quite sure what to do from here.

Any help would be greatly appreciated,

Thank you.

Answer 1

Let $M < 0$ be given, choose N such that $N < \sqrt[3]{M-1}$. For $x < N$, we have: $1 + x^3 < 1 + (M - 1) = M$. This proves the result.