Does there exist a real valued differentiable function such that $\lim_{n \rightarrow \infty} f(x) = 2$ and $\lim_{ n\rightarrow \infty} f'(x) =1$?

Question

This question appeared in my exam (in my university entrance test).

Question: Does there exist a real valued differentiable function such that $\lim_{x\rightarrow \infty} f(x) = 2$ and $\lim_{x\rightarrow \infty} f'(x) =1$?

My attempt: $2=\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\frac{e^xf(x)}{e^x}=\lim_{x\to\infty}(f(x)+f'(x))=2+1=3$. Hence contradiction!

Am I correct?

$\textbf{Has anyone seen this problem before in college contests, problem book etc.?}$ If you have seen this problem before please let me know.

Nice, but not new: https://math.stackexchange.com/a/42848/42969 — Martin R, Jul 21 '22 at 15:01

Hyperbolic PDE friend · Answer 1 · 2022-07-21T15:28:49.587

2

If $\lim_{x \rightarrow \infty} f'(x) = 1$, then there is some $C > 0$ such that $f'(x) > \frac{1}{2}$ for all $x \geq C$. It follows that $$ \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} f(C) + \int^x_C f'(s)~\mathrm{d}s \geq f(C) + \int^x_C \frac{1}{2}~\mathrm{d}s = f(C) + \frac{x-C}{2} \overset{x \rightarrow \infty}{\longrightarrow} \infty. $$ So such function can't exist.

edited Jul 21 '22 at 15:28

answered Jul 21 '22 at 14:59

Hyperbolic PDE friend

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As above, if a function goes to a fixed value, it's derivative must eventually approach 0. A derivative of 1 means it keeps increasing – Barry Carter Jul 21 '22 at 15:00
He used l'Hospital rule. – hamam_Abdallah Jul 21 '22 at 15:00
Ah, I see, thank you. His seems correct at first sight, although I do not know whether l'Hopital can be used if the fraction is not shortened. Guess it can't, since his limit cannot be correct. – Hyperbolic PDE friend Jul 21 '22 at 15:02
Unless I am terribly mistaken, the proof in the question is correct, and the question is a duplicate. – Martin R Jul 21 '22 at 15:04
Might be true... – Hyperbolic PDE friend Jul 21 '22 at 15:05
by changing $0$ to $C$ you have $\frac{(x-C)}{2}$ too. – zwim Jul 21 '22 at 15:07
ofc, thank you! – Hyperbolic PDE friend Jul 21 '22 at 15:28

Does there exist a real valued differentiable function such that $\lim_{n \rightarrow \infty} f(x) = 2$ and $\lim_{ n\rightarrow \infty} f'(x) =1$?

1 Answers1