 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/4/a/solution

The cancellation of the quadratic terms makes the fourth-order terms decisive. The Taylor expansions give

1 + x^{2} - e^{x^{2}} = - \frac{x ^{4}}{2} - \frac{x ^{6}}{6} + O (x^{8}), x sin x - x^{2} = - \frac{x ^{4}}{6} + \frac{x ^{6}}{120} + O (x^{8}) .

Canceling

x^{4}

for

x \neq = 0

and taking the limit yields

x \to 0 lim \frac{1 + x ^{2} - e ^{x^{2}}}{x sin x - x ^{2}} = 3.

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