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$
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$
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.