What is $-\infty \cdot - \infty$?

Question

What is $-\infty \cdot - \infty$?

And what do I do if I come across it in an assignment when taking the limit?

E.g. $$\lim_{x \to - \infty} (x \cdot x) $$

Since $x^{2}$ is continuous, can you just set in for $x$ and get:

$$\lim_{x \to - \infty} (x^{2}) = \infty$$

Answer 1

For any $x$,

$$x\cdot x=(-x)(-x)$$ so anything you can conclude for $\infty$ is also valid for $-\infty$.

Answer 2

Though this has been answered in the comment, I am writing a proof for this so as to

1. clear this question from the unanswered queue, and
2. provide a direct argument from the basics, independent of the product rule for continuous function because to Prove that the product of two continuous functions is continuous, you need the limit values $f(a)$ and $g(a)$ to be finite and $a \in \Bbb{R}$ as $x \to a$ so that $f(a)$ and $g(a)$ are properly defined.

Let $M > 1$ be an arbitrarily large number. For all $x < -M$, $x^2 > M^2 > M$, so that $x^2 \to \infty$ as $x \to -\infty$.