Answer
Definition: Let $f$ be a function defined on some interval $(-\infty, a)$. Then
$\lim\limits_{x \to -\infty} f(x) = -\infty$ means that for every negative number $M$ there is a corresponding negative number $N$ such that if $x \lt N$ then $f(x) \lt M$
$\lim\limits_{x \to -\infty}(1+x^3) = -\infty$
Work Step by Step
Definition: Let $f$ be a function defined on some interval $(-\infty, a)$. Then
$\lim\limits_{x \to -\infty} f(x) = -\infty$ means that for every negative number $M$ there is a corresponding negative number $N$ such that if $x \lt N$ then $f(x) \lt M$
Let $f(x) =(1+x^3)$
This function is defined on the interval $(-\infty,\infty)$
Let $M \lt 0$ be given.
Let $N = min\{-2, M\}$
Suppose that $x \lt N$
Then:
$(1+x^3) \lt x \lt N \leq M$
Therefore, $\lim\limits_{x \to -\infty}(1+x^3) = -\infty$
