Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 19 / a / Solution

The Big O notation means that there exist constants $C>0$ and $\delta >0$ such thatwithin the domain of $f$. Thus $f(x)/x^2$ stays bounded in a punctured neighbourhood of zero. It need not have a limit: $f(x)=x^2\sin (1/x)$ is one example. The statement imposes no condition on a separately assigned value $f(0)$.