If $f(x)\rightarrow L$ from both sides then $f'(c)=0$ for some $c$ Let $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ is differentiable over $\mathbb{R}$.

Prove that if $\underset{x\rightarrow\infty}{\lim}f(x)=L$ and $\underset{x\rightarrow-\infty}{\lim}f(x)=L$ for $0<L\in\mathbb{R}$, then $\exists c\in\mathbb{R} \; f'(c)=0$.

Thoughts: I know that if the interval is $[a,b]$ (where $a, b \in \mathbb{R}$) $x\rightarrow a^+$ and $x\rightarrow b^-$, I can use Rolle's theorem, but what should I do when $x$ approaches infinity?

Without loss of generality we can assume $f(0)<L$. There exist two points $x_1<0$ and $x_2>0$ such that $f(x_1)=f(x_2)=\frac{1}{2}\bigl(f(0)+L\bigr)$. Therefore there exists some $\xi\in(x_1,x_2)$ such that $f'(\xi)=0$.
