Let $f$ be a continuously differentiable function and let $L=\lim_{x\to\infty}(f(x)+f'(x))$ be finite.
Does this imply that if $$\lim_{x\to\infty} f'(x)$$ exists, then it is equal to $0$?
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See this. – David Mitra Dec 22 '14 at 12:35
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You can see here as well ! – sciona Dec 22 '14 at 12:40
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Yes. This is because if $f'(x) > C>0$ for all $x>M$, then $f(x)$ goes to $\infty$ as $x\to\infty$.
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1Also it appears that the extra hypothesis that $f'$ must have a limit as $x\to\infty$ can be dropped, since one can prove that from the fact that $f+f'$ has the limit $L$ (see linked thread). – Jeppe Stig Nielsen Dec 22 '14 at 12:58