If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h

[kbgregory]

Homework Statement

If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h

Homework Equations

We know that f'(a) = lim[h->0] (f(a+h)-f(a))/h

The Attempt at a Solution

I have done the following, and I am not sure if it is correct, though the result makes sense intuitively:

lim[h->0] (f(a+2h)-f(a+3h))/h

= 2* lim[h->0] (f(a+2h)-f(a+3h))/ (2*h)

= 2* lim[h->0] (f(a+2h)-f(a)-f(a+3h)+f(a))/ (2*h)

= 2* { lim[h->0] (f(a+2h)-f(a))/(2*h) } - 2*{lim[h->0] f(a+3h)-f(a))/ (2*h)

And here is part about which I am unsure, since I am working with multiples of h:

= 2*f'(a) - 3*{lim[h->0] f(a+3h)-f(a))/ 3*h)

= 2*f'(a) - 3*f'(a) = -f'(a)

 

[Dick]

Looks fine to me.