Suppose f' is continuous. Show that lim h -> 0 (f(x+h)-f(x-h))/2h=f'(x)

Question

This problem is for the limit as h approaches 0.

Answer 1

Let's add zero to numerator -f(x)+f(x)=0

now equation is (1/2)*[(f(x+h)-f(x))+(f(x)-f(x-h))]/h as h approaches zero

h is a common denominator

so equation becomes (1/2)*(f'(x)+f'(x))=f'(x)

(f(x)-f(x-h)/h as h approaches zero and h>0 is f'(x) evaluated as f(x-h) approaches f(x) on left side of x-axis