Please some one help me to do the following problem.
If $f : \mathbb R\to\mathbb R$ is differentiable at $0$ and $f'(0) =1$, find $\lim\limits_{x\to 0} \frac{f(x) - f(-x)}x$.
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5Write it as $$\frac{f(x) - f(0)}{x} + \frac{f(-x) - f(0)}{-x} $$ – Prahlad Vaidyanathan Dec 14 '13 at 05:10
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1$2f'(0)$ try to find out how. – toufik_kh.17 Dec 14 '13 at 05:14
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Thanks ..But when you put -f(0) for the first part, in the second part it should be f(0) right.. Then how do we find the value of the second part. – Chans Dec 14 '13 at 05:16
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1a change of variable will do the job in the second part. – toufik_kh.17 Dec 14 '13 at 05:20
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Ok, I got it. thanks a lot.. – Chans Dec 14 '13 at 05:23
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$\displaystyle \lim_{x \to 0} \frac{f(x) - f(0)}{x} = f'(0)$
$\displaystyle \lim_{x \to 0} \frac{f(-x) - f(0)}{-x}=$$\displaystyle \lim_{y \to 0}\frac{f(y) - f(0)}{y}=f^{'}(0)$
toufik_kh.17
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Thank you very much. Some how I was trying this problem in many ways. I understood it well now. – Chans Dec 14 '13 at 05:34