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Limit inferior of functions at infinity

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How do you define $$\liminf_{x\to\infty} f(x) where $a>0$, and $f$ is continuous on $\mathbb R$?

Is it the following?

There exist $x_{0}$ such that $f(x) for all $x>x_{0}$

calculus real-analysis limsup-and-liminf
asked 2012-08-12
user id:37754
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    The lhs of your inequality is either a real number $z$ or $-\infty$. In the first case the inequality is true if $z< a$. In the second case it is automatically true (since you assume $a>0$). (And no, your approach won't work). – 2012-08-12

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