# Prove if $\lim_{x\to +\infty}(f(x)+f'(x))=0$ then $\lim_{x\to +\infty} f(x)=0$ [duplicate]

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

• Have you tried to solve the differential equation $f'(x) = -f(x)+ g(x)$, where $g(x)$ is a function, that goes towards $0$ for large $x$? Apr 19, 2020 at 20:19
• @HamidMohammad No i did not {}{}{}{} Apr 19, 2020 at 20:20
• Thanks a lot a lot a lot. Apr 19, 2020 at 20:57

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.
• What if $f’$ has no limit? Apr 19, 2020 at 21:05