Lipschitz functions are $o(|x|)$?

Question

Consider a Lipschitz function from $\mathbb{R}\to\mathbb{R}$. Can we say that $\lim_{x\to\infty}\frac{f(x)}{|x|}=0$. Can we also say that $f$ is differentiable. Continuity is quite evident. But linear order is hard to come by. Any hints? Thanks beforehand.

Answer 1

No, consider the function $f(x)=|x|$ which is Lipschitz in $\mathbb{R}$ (with constant $1$), not differentiable at $0$, and $$\lim_{x\to\pm\infty}\frac{f(x)}{|x|}=\pm 1.$$

Answer 2

$f(x)=x$ is a counterexample for the first part. Lipschitz functions need not be differentiable at every point but they are differentiable almost everywhere.