When Lim h tends to zero, (f (x0 + H) - f (x0-h)) / 2H = f '(x0) can't understand

When Lim h tends to zero, (f (x0 + H) - f (x0-h)) / 2H = f '(x0) can't understand

(f(x0+h)-f(x0-h))/2h=(f(x0+h)-f(x0)+f(x0)-f(x0-h))/2h=1/2 * ( ((fx0+h)-f(x0))/h + ((fx0-h)-f(x0))/(-h) )=1/2 ( f'(x0) + f'(x0) )= f'(x0)