Past exam of the mathematical methods II course of the University of Oxford / 2015 / 2 / b / Solution

Combine the terms before taking their limit:Using the exponential function Maclaurin series, the numerator is $x^2/2+O(x^3)$ and the denominator is $x^2+O(x^3)$. ConsequentlyThe cancellation removes the two individual singularities. Subtracting their separate divergent limits would not determine the finite answer.