Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 2 / 9 / i

First expand the exponential function and divide by $x$:Seek its reciprocal as $1+Ax+B{x}^2+C{x}^3+O(x^4)$. Equating coefficients in the product with this expression givesHence $A=1/2$, $B=1/12$, $C=0$. The first three nonzero terms of the Taylor expansion areAlthough the quotient is undefined at zero as written, its limit is one, and this Taylor expansion belongs to its continuous, real analytic extension.