How should I use the alternate formula of the derivative to “mathematically” show that the derivative of $f(x)=|x-2|$ at $x=2$ does not exist?

Question

The title is the question. But I’m not sure how one can “mathematically” show the derivative? Can someone please interpret that part of the problem? Thanks!!

Answer 1

Generally, the alternate definition of the derivative is

$$f'(c)=\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}$$

Note that this definition requires a limit. Thus we must have that

$$\lim_{x\rightarrow c^{\color{red}{-}}}\frac{f(x)-f(c)}{x-c}=\lim_{x\rightarrow c^{\color{red}{+}}}\frac{f(x)-f(c)}{x-c}$$

Take left and right limits using the definition here. What happens?

Answer 2

In general, you should draw the graph of the one variable function. Then take the right and left limit at the "corner" point of that graph, which gives you different values. In your function $f$, it happens at $x=2$.