Question

f*(x) = lim as h -> 0 [f(x+h)-f(x-h)]/ h

write an equation that expresses the relationship between the functions f*(x) and f`

I know that f` is [f(x+h)-f(x)]/h but i have no clue how to write an equation to relate the two!!!

Answer 1

f(x+h)-f(x-h) =

f(x+h) - f(x) + f(x) - f(x-h)

lim as h -> 0 [f(x+h)-f(x-h)]/ h =


lim as h -> 0 [f(x+h)-f(x)]/ h +

lim as h -> 0 [f(x)-f(x-h)]/ h

lim as h -> 0 [f(x)-f(x-h)]/ h

also equals the derivative of f if the functon is differentiable.