Limit of a periodic function

Let $f:\R \rightarrow \R$ be a periodic function with period $T>0$ , such that $f$ admits a limit as $x\rightarrow+\infty$ .
What can be said about $f$ ?

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Solution by Ancient Tree

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Let $x\in \R$ be an arbitrary real number, and consider the sequence $f(x),f(x+T),f(x+2T),\ldots$ . By assumption, since $x+n T \underset{n \rightarrow+\infty}{\longrightarrow}+\infty$ , it converges to a limit $l\in \R$ . But this sequence is constant by periodicity of $f$ . This means that $f(x)=l$ . Applying the same reasoning for every $x\in \R$ , we get that $f$ is constant, equal to $l$ .

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