f(y)= 1/y

f(0)=0

rule

F(x)= 1/[x？ sin( 1/x)]

F(0)=0

In this example, the derivative f'(0) does not exist.

It is wrong to simply say that the limit does not exist. This limit may or may not exist, depending on the definition of f(0).

The first floor is not understood correctly. Although the limit of sin( 1/x) does not exist, but x? Sin( 1/x)→0, no problem.