On the mere definition of divergence

Question

If we have $$\lim_{x\to\infty}f(x)=\infty$$ we say that as x tends towards infinity $f(x)$ diverges.

For what reason do we not say

as x tends towards infinity $f(x)$ converges to $\infty$?

Answer 1

You can say that it converges in the extended real numbers, sometimes denoted $\overline{\mathbb{R}}$. However, usually in the context of the ordinary real numbers we do not say that it converges simply because $\infty$ is not a real number.