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Let $f:[0,\infty)\to\mathbb{R}$ be continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ suppose: $$\lim_{x\to\infty}{f(x)+f'(x)}=5$$ Show that $f$ is uniformly continuous.

I've thought of proving that $\lim_{x\to\infty}{f(x)}$ is finite but I'm not even sure if $f$ converges at infinity.

Another idea I thought of is proving the Lipschitz inequality for $f$ or to just show that $f'$ is bounded but again I can't see how to do it without assuming that $f$ converges.

Can I get a hint or a direction in order to solve this?

pompabalompa
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    See if working with $e^{-x}(e^x f(x))’$ gets you anything. – A rural reader Mar 02 '24 at 22:02
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    That $f(x) \to 5$ under these circumstances is a famous problem. See e.g. https://math.stackexchange.com/questions/840957/how-can-this-be-proved-lim-x-to-inftyfxfx-l or https://math.stackexchange.com/questions/407654/if-lim-limits-x-rightarrow-infty-fxfx-l-infty-does-lim-limits (if $f(x) + f'(x) \to L$ as $x \to \infty$ then $f(x) \to L$ as $x \to \infty$) and https://math.stackexchange.com/questions/2003989/if-f-is-continuous-and-with-a-limit-at-infinity-then-f-is-uniformly-continuo (if $f: [a, \infty) \to \mathbb{R}$ is continuous and has a limit at infinity then $f$ is uniformly continuous) – leslie townes Mar 02 '24 at 22:17

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