TITLE: Calculating Limit $ \lim\limits_{x\to 0} \frac{\mathrm d^2}{\mathrm dx^2} \frac{f(x)}{x} $
QUESTION [5 upvotes]: 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!

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

REPLY [8 votes]: 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 $f'''(0)/3$.