Is a function differentiable in $x$ if $\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\infty$? All the definitions of differentiability I found (Wolfram Mathworld for instance) only require this limit to exist, but say nothing about the domain in which that has to happen.  
So what if that limit is $\pm\infty$? Wouldn't (in the "normal" interpretation of a function $\mathbb R \rightarrow \mathbb R$, for instance) the slope of the tangent of $f$ in $x$ be undefined?
Edit: a link clearing it up.

This is an unfortunate case of terminology getting the better of us. The definition of a limit is

The limit of the function $f(x)$ as $x$ approaches $a$ exists if there is a number $L$ such that for all $\epsilon>0$, there exists a $\delta>0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$. In this case, we say that $$\lim_{x\to a} f(x)=L.$$

So the limit, if it exists is a number. Infinity is not a number, so technically, if $\lim_{x\to a}f(x)=\infty$, the limit doesn't exist. And yet it is infinity. It's confusing at first, but once you get used to it, it's not too bad.
