Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 1

The Maclaurin series of the exponential function and sine giveBoth the numerator and denominator first become nonzero at order $x^4$. Dividing by that common power gives the limit

The fundamental theorem of calculus differentiates the variable upper endpoint:At zero, $I(0)=0$, $I^{\prime }(0)=2/\sqrt{\pi }$, $I^{\prime \prime }(0)=0$ and $I^{\prime \prime \prime }(0)=-4/\sqrt{\pi }$. The integrand is even, so its integral from zero is an odd function; all even coefficients in the Maclaurin series vanish. HenceThis is the local expansion of the error function.