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- Apr 13, 2013

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I am looking at this exercise:

Let f: (-a,a)[tex]\rightarrow \mathbb{R}[/tex] continuous at [tex]0[/tex].We suppose that for a [tex]t \epsilon (0,1) [/tex] this condition is satisfied:

[tex] \lim_{x\rightarrow 0}\frac{f(x)-f(tx)}{x}=b [/tex]

Show that [tex] f'(0) [/tex] exists and find its value.

I thought that I could write it like that [tex] \lim_{x\rightarrow 0}\frac{f(x)-f(0)+f(0)-f(tx)}{x}=b \Rightarrow \lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x}+\lim_{x\rightarrow 0}\frac{f(0)-f(tx)}{x}=b\Rightarrow (1-\frac{1}{t})\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x}=b[/tex]

So,[tex]f'(0)[/tex] exists and is equal to [tex] \frac{bt}{t-1} [/tex] .Are my thoughts right or am I wrong?