Can we say $\lim_{x \to 0} f(x) = \infty$, where $f(x)$ is defined as, for example :
$$ f(x) := \begin{cases} \frac{1}{x} & x > 0 \\ - \frac{2}{x} & x < 0 \end{cases} $$
Both limits $x \to 0+$ and $x \to 0-$ diverge to $+ \infty$ according to the $\varepsilon-M$ definition of a limit of a function, so it seems we can say that the whole limit is $+ \infty$. But what I'm concerned is that the speed of divergence differs whether $x > 0$ or $x < 0$ - a similar situation can be found in the improper integral $\int_{\mathbb{R}} x dx$. One might say the value of the integral is $0$ but the speed of divergence might differ so the answer is undefined, unless we're talking about Cauchy Principal values.
But the limit here is not an integral and both limits go to $+ \infty$ so I want to know if we can say $\lim_{x \to 0} f(x) = + \infty$ here.