Limit of derivative function at infinity 
$f$ is a differentiable function on real line such that $$\lim_{x \to \infty}f(x)=1$$ and $$\lim_{x \to \infty}f'(x)=\alpha$$. Then what can be said about $\alpha$

*

*$\alpha=0$

*$\alpha$ may not be $0$ but $|\alpha| \le1$

*$\alpha\geq1$

*$\alpha\leq-1$

I cannot think of any viable option other than 1,but i cannot prove it. Any hint or solution would be appreciated!

Because $\lim_{x \to \infty}f(x)=1$, $y=1$ is a horizontal asymptote of the graph of $f(x)$.
Therefore, $\lim_{x \to \infty}f'(x)=0$.
