Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 12 / a / ii / Solution

Although $\cos (1/x)$ oscillates without a limit, its magnitude is at most one. Therefore $|x\cos (1/x)|\le |x|\to 0$. The squeeze theorem givesThe shrinking factor $x$ suppresses the oscillation from either side of zero.