real analysis - Help proving that $lim_{x rightarrow infty} f(x)=L$ and that $lim_{x rightarrow infty}f'(x)=0$

I am trying to prove that if $f$ is a differentiable function on some $(c,\infty)$ and supposing that $\lim_{\rightarrow \infty} [f(x)+f'(x)]=L$, where L is finite, then $\lim_{x \rightarrow \infty} f(x)=L$ and that $\lim_{x \rightarrow \infty}f'(x)=0$. The hint the book gives is to set $f(x)=\frac{f(x)*e^x}{e^x}$, but I don't see how it could be useful.

Thanks for your help!