$\lim_{|x| \to \infty} f(x)=0$ meaning

Question

I was wondering if this limit $\lim_{|x| \to \infty} f(x)=0$ is correct to be interpreted as:

$$\lim_{x \to \infty} f(x)=0 \qquad\text{and}\qquad \lim_{x \to -\infty} f(x)=0?$$

Is this correct?

Answer 1

For $x \in \mathbb{R}$, yes. In general, it would be interpreted as "for every sequence $(x_n) \in \mathbb{K}^n$, $|x_n| \rightarrow \infty$ implies $f(x_n) \rightarrow 0$.