Why is a function at sharp point not differentiable?

I am learning about differentiability of functions and came to know that a function at sharp point is not differentiable.
For eg. $$f(x)=|x|$$ I could find out that $f(x)$ is not differentiable at $x=0$ because
$$\lim_{x\to 0^-}f'(x) \ne \lim_{x\to 0^+}f'(x) $$ This is all mathematical but I couldn't understand where the sharp point plays its role here ?
How sharp point makes these limits to evaluate different ?

Solution 1:

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Solution 2:

First remark: your $f$ is not differentiable (at $0$) because the limit $$ \lim_{h \to 0} \frac{|h|}{h} $$ does not exist. In general the limit of $f'$ is only a sufficient condition for differentiability. Be very careful, if you use it to disprove differentiability.

Have you tried to sketch the graph of $f$? If so, you have seen that there is no tangent line to the graph at $0$, because of the sharp point. This is way to "understand" the rôle of the sharp point. But again, be careful: differentiability is a mathematical idea. The best way to understand it, is to understand it mathematically, according to the definition. Everything else may be misleading.