If f is a continuous function, what is the limit as h approaches 0 of the average value of f on the interval [x; x + h]

Question

 If f is a continuous function, what is the limit as h approaches 0 of the average value of f on the interval [x; x + h] 

Answer 1

The average value of y = f(x) from x to x + h is

 

[f(x + h) - f(x)]/[(x + h) - x] = [f(x + h) - f(x)]/h

 

Limit as h→0 [f(x + h) - f(x)]/h = f'(x)

 

 

 

 

Answer 2

Your interval is (a,b) ,  where a=x   and   b = (x+h)

 

 
limit     [f(b)- f(a)] / (b-a)
 h→0

 

 

limit     [f(x+h) - f(x)] /(x+h-x)

h → 0

 

limit     [f(x+h) - f(x)] /h
h→0