Limit similar to differentiation

[Diffy]

Homework Statement

f a differentiable real valued function

lim h - > 0 of (f(x + ah) - f(x + bh))/h

where a,b real numbers

Homework Equations

definition of derivative

lim h-> 0 of (f(x+h) - f(x))/h

The Attempt at a Solution

I've picked several functions like x^2 and 1/2x

in the first case you get 2xa - 2xb = 2x(a-b)

in the latter you get 1/2a - 1/2b = 1/2(a-b)

This leads me to suspect f'(x)*(a - b) as the solution, I just have no idea what to do to the limit.

I tried working backwards from my solution, but I still am missing something. Any help is appreciated.

 

[statdad]

See if this gets you anywhere.

$$\frac{f(x+ah)-f(x+bh)}{h} = \frac{f(x+ah)-f(x)}{h} + \frac{f(x) - f(x+bh)}{h} = a\left(\frac{f(x+ah)-f(x)}{ah}\right) - b \left(\frac{f(x) - f(x+bh)}{bh}\right)$$

 

[Diffy]

thanks, that last one since you are multiplying by -b the numerator should be f(x +bh) - f(x)

It is clear that as h -> 0 ah and bh ->0 and therefore those we are left with f'(x)

Thank you so much for your help.