Derivative is the limit of a special format (that is, the limit of the ratio of the dependent variable to the independent variable of the function y=f(x) when the independent variable tends to 0), but the general limit is not derivative;

The left and right derivatives are the limits of the above scheme when the independent variable tends to zero from one side. Similarly, the general left and right limits are not left and right derivatives.

In short, the derivative is the limit, and the limit is not necessarily the derivative.