Let $f$ be differentiable on $(0,\infty)$.Show that $\lim\limits_{x\to \infty}(f(x)+f'(x))=0$,then $\lim\limits_{x\to \infty}f(x)=0$

Let $f$ be differentiable on $(0,\infty)$. Show that $\lim\limits_{x\to \infty}(f(x)+f'(x))=0$, then $\lim\limits_{x\to \infty}f(x)=0$

My attempt:

When both $f(x)$ and $f'(x)$ $\to 0$ when $x\to \infty$ then the problem is trivial. But, the problem is I cannot do anything more than the trivial case. I definitely realise that if $\lim\limits_{x\to \infty}f(x)=0$ then $\lim\limits_{x\to \infty}(f(x)+f'(x))=0$ is true. But, I cannot really prove the converse. Please help. Thank you.

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