How to calculate $\liminf$ of the function $f(x)=1-x^2$ Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ given by
$$f(x)= 1-x^2, \; \forall \; x \in \mathbb{R}.$$
Question. How to evaluate
$$\liminf_{|x|\rightarrow \infty} f(x)?$$
What I got to think about is
$$\liminf_{|x|\rightarrow \infty} f(x)=\min\{c \in \mathbb{R} \; ; \; x_n \rightarrow \infty \: \text{and} \: f(x_n)=c\}.$$
It's true? Or is my reasoning wrong?
The function $f(x)=1-x^2$ is not bounded at infinity, that is, for all $A&gt;0$ and $k&gt;0$  there exists $x_0 \in \mathbb{R}$ such that $x_0&gt;A$ we have $|f(x_0)|&gt;k$ $(\lim_{x \to \infty} 1-x^2=-\infty)$.
Since $$\liminf_{x \rightarrow \infty} f(x)=\sup_{N&gt;0}\inf_{x \geq N} f(x)$$ and for all $N&gt;0$ we have $$\inf_{x \geq N} f(x)=-\infty$$ then $$\liminf_{x \rightarrow \infty}(1-x^2)=-\infty.$$
Similarly, we can handle the other case.
$$\lim_{|x| \rightarrow +\infty} f(x) = -\infty$$
so $$\liminf_{|x| \rightarrow +\infty} f(x) = \limsup_{|x| \rightarrow +\infty} f(x) =-\infty$$