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

Using the standard sine limit $\sin t/t\to 1$, with angles in radians,Thus the two-sided limit exists and is finite.

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.

The derivative of the exponential function at zero gives $e^{-x}=1-x+O(x^2)$. Hence $1-e^{-x}=x+O(x^2)$, andThe denominator is nonzero for every real $x\ne 0$, so this is a two-sided finite limit.