Why Absolute Value Function |x| is Not Differentiable at x=0

Question

The definition of a derivative is the slope of a function tangent to a point. It is also defined as $$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$
If we apply this to $f(x)= |x|$, we get that it is $\lim\limits_{h\to 0} \dfrac{|h|}{h}$, which is undefined. However, if we look at the graph of $|x|$, we see that there can exist a tangent line at x=0, with slope 0. So why is the derivative undefined instead of 0?

Answer 1

Let $f(x) = |x|$. For $x > 0$, the gradient is $1$, whereas for $x < 0$, the gradient is $-1$. At $x = 0$, you have infinitely many lines, which are represented by the subgradient $[-1,1]$.