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Today I had an exam and the following problem came up. I have absolutely no idea how to approach this. Any help in solving this is appreciated!

$ \lim_{x\to 0} \frac{\mathrm d^2}{\mathrm dx^2} \frac{f(x)}{x},\qquad f(0) = 0$

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    @anon: Oh, I understood my mistake. When differentiating the Tailor series with respe$c$t to x I also differentiated the f'(0), f''(0) and f'''(0). Which is wrong, since those are just constants.2011-09-06

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The second derivative of $f(x)/x$ can be found with two applications of the quotient rule: \frac{x^2f''(x)-2xf'(x)+2f(x)}{x^3}. Now to evaluate the limit of this as $x\to0$ we can take Iasafro's suggestion from the comments of using a trick called L'Hôpital's rule. Taking the derivative of numerator and denominator above leads to a lot of cancelling terms, which comes out to be \frac{f'''(x)}{3}. Taking the limit gives f'''(0)/3.