What can we say about $\alpha$ if $\lim_{x\to \infty}f(x)=1$ and $\lim_{x\to \infty}f'(x)=\alpha$ for a differentiable function $f$ on $\mathbb{R}$?

Question

For what values of $\alpha$ would the given conditions hold? I don't know how to proceed.

What are your thoughts on the question? Do you have a guess? — Ben Grossmann, Jun 11 '16 at 10:05
Apply MVT to $f(x+1)-f(x)$. What happens as $x\rightarrow\infty$? — Wojowu, Jun 11 '16 at 10:10
Similar: http://math.stackexchange.com/questions/42277/limit-of-the-derivative-of-a-function-as-x-goes-to-infinity — , Jun 11 '16 at 10:11

score 2 · Accepted Answer · answered Jun 11 '16 at 10:11

If $\lim_{x\to \infty}f(x)=a$ and $\lim_{x\to \infty}f'(x)=b$, then $b=0$.

Hint:

Show if $\lim_{x\to \infty}f'(x)=b$, then for any given $h>0$, $\lim_{x\to \infty} \dfrac{f(x+h)-f(x)}{h}=b$.
Show if in addition we have $\lim_{x\to \infty}f(x)=a$, then $b=0$.

Then two steps follows from MVT and basic definition of limit.

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