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


Imagine two semicircles joining when $x=x_0$ such as 
$$f(x) = \sqrt{1-(x_0+1-x)^2} \text{ when } x_0 \le x \le x_0+2$$
$$f(x) = -\sqrt{1-(x_0-1-x)^2} \text{ when } x_0-2 \le x \le x_0$$
Change the signs if you want alternative limits for the derivatives as $x \to x_0$
