Prove if $\lim_{x\to +\infty}(f(x)+f'(x))=0$ then $\lim_{x\to +\infty} f(x)=0$ Let $f$ be a real function  continuously differentiable at $\Bbb R$ such that
$$\lim_{x\to +\infty}(f(x)+f'(x))=0$$
prove that
$$\lim_{x\to +\infty} f(x)=0$$
I tried tu use exponential function knowing that
$$\frac{d}{dx}f(x)e^x=(f(x)+f'(x))e^x$$
but I got nothing.
thanks in advance for an answer or un idea

Could you prove it by contradiction? If $\lim_{x\rightarrow \infty} f'(x) \neq 0$ then $\lim_{x\rightarrow \infty} f(x) = -\lim_{x\rightarrow \infty} f'(x)$ but if $f'>0$ then $f$ is increasing, and if $f'<0$ then $f$ is decreasing - that seems like a contradiction.
