The limit of the ratio of the vertical increment to the horizontal increment when x tends to 0 is equal to the derivative of the function.
let f(x) be a function that is defined and differentiable at a point x=a. Then the derivative of f(x) at x=a, denoted by f'(a), is defined as the limit of the ratio of the change in the function value to the change in the input value as the latter change approaches zero:
f'(a) = lim (f(x) - f(a))/(x - a) x→a x→a