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Seemingly Simple Derivative (as a limit) Problem

by luke8ball
Tags: derivative, limit, seemingly, simple
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luke8ball
#1
Nov20-12, 02:01 PM
P: 22
I'm having trouble showing the following:

lim [f(ax)-f(bx)]/x = f'(0)(a-b)
x→0

I feel like this should be really easy, but am I missing something? I tried to use the definition of the derivative, but I know I can't just say f(ax)-f(bx) = (a-b)f(x).

Any ideas?
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arildno
#2
Nov20-12, 02:04 PM
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Try to add zero in your numerator in the shape f(0)-f(0), and see if you can rearrange it in a clever manner.
luke8ball
#3
Nov20-12, 02:11 PM
P: 22
You mean so that I get:

[lim f(ax) - f(0)]/x - [lim f(bx) - f(0)]/x
x→0 x→0

I had thought about that, but I still don't see how that gives me af'(0) - bf'(0)...

arildno
#4
Nov20-12, 02:20 PM
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Seemingly Simple Derivative (as a limit) Problem

Think chain rule..
arildno
#5
Nov20-12, 03:05 PM
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A further hint:
Let g(x)=ax. Then, g(0)=0


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