Limit of derivative does not exist, while limit of difference quotient is infinite Can anyone show an example of a function $f$ of a real variabile such that


*

*$f$ is differentiable on a neighborhood of a point $x_0 \in \mathbb{R}$, except at $x_0$ itself;

*$f$ is continuous at $x_0$;

*$\displaystyle \lim_{x \to x_0} f'(x)$ does not exist;

*$\displaystyle \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = +\infty$ or $\displaystyle \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = -\infty$ ?


What about 
$$f(x)=\sqrt{x}+x\sin\frac{1}{x}$$
defined on $(0,+\infty)$, that can be defined by continuity at $0$ with $f(0)=0$? (I let you check that it works, if I am not wrong)
If you don't like it because it is only $\mathbb{R}_+$, you can replace with
$$f(x)=\sqrt[3]{x}+x\sin\frac{1}{x},$$
graph
that can be defined on $\mathbb{R}$.
Using google, you get a fairly obvious drawing explaining what is happening (personally before having an explicit example, I had a fairly good drawing in my head to convince myself that such an example was possible, and then help me to produce a formula)
