 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/15/c/ii/solution

Both numerator and denominator vanish quadratically. Factor out

x^{2}

before dividing the power series:

\frac{x s i n x}{l n ( 1 + x ^{2} )} = \frac{1 - x ^{2} /6 + x ^{4} /120 + O ( x ^{6} )}{1 - x ^{2} /2 + x ^{4} /3 + O ( x ^{6} )} .

The reciprocal of the denominator is

1 + x^{2} /2 - x^{4} /12 + O (x^{6})

. Hence

\frac{x sin x}{ln ( 1 + x ^{2} )} = 1 + \frac{x ^{2}}{3} - \frac{19 x ^{4}}{120} + O (x^{6}) .

The expansion describes the continuous extension at the removable singularity

x = 0

, where the original quotient is undefined but its limit is one. Odd powers vanish because the quotient is even.

 New to topics? Read the docs here!