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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?

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The function $f(x)=1-x^2$ is not bounded at infinity, that is, for all $A>0$ and $k>0$ there exists $x_0 \in \mathbb{R}$ such that $x_0>A$ we have $|f(x_0)|>k$ $(\lim_{x \to \infty} 1-x^2=-\infty)$.

Since $$\liminf_{x \rightarrow \infty} f(x)=\sup_{N>0}\inf_{x \geq N} f(x)$$ and for all $N>0$ 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.

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$$\lim_{|x| \rightarrow +\infty} f(x) = -\infty$$

so $$\liminf_{|x| \rightarrow +\infty} f(x) = \limsup_{|x| \rightarrow +\infty} f(x) =-\infty$$

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  • $\begingroup$ Could you elaborate better, please? $\endgroup$ – Guilherme de Loreno Sep 16 at 16:29

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