If $\lim\limits_{x \to \pm\infty}f(x)=0$, does it imply that $\lim\limits_{x \to \pm\infty}f'(x)=0$?

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and

1. $\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$,
2. there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$.

Can we say anything about $\lim_{x \to \infty}f'(x)$ and $\lim_{x \to -\infty}f'(x)$?

I am tempted to say that $\lim_{x \to \infty}f'(x)$ = $\lim_{x \to -\infty}f'(x)=0$.

I started with the following, but I'm not sure this is the correct approach, $$\lim_{x \to \infty}f'(x)= \lim_{x \to \infty}\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}.$$