# If $\lim_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists, what about $\lim_{x\to\infty}f(x)$? [duplicate]

Given that $f(x)$ is continuous on $[0,\infty]$.

If $\lim\limits_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists

then evaluate $\lim\limits_{x\to\infty}f(x)$

• This and this may help. (Take $G(x)=\int_0^x f(t)\,dt$ and note $f=G'$.) – David Mitra Jun 24 '15 at 12:20
• It's similar to Barbalat’s Lemma! – Reza H. Khayyami Jun 24 '15 at 12:34