If $\lim_{x \rightarrow \infty} f(x) + f'(x) = 0$, prove $\lim_{x\to \infty} f(x)$ and $\lim_{x \rightarrow \infty} f'(x)$ exist and both equal $0$.
If $\lim_{x\to\infty}(f(x)+f'(x))=L$ show that $\lim_{x\to\infty} f(x) = L$ and $\lim_{x\to\infty} f'(x) = 0$