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

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Solution

The Maclaurin series of the exponential function and sine give

e^{x^{2}} = 1 + 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}) .

Both the numerator and denominator first become nonzero at order

x^{4}

. Dividing by that common power gives the limit

x \to 0 lim \frac{1 + x ^{2} - e ^{x^{2}}}{x sin x - x ^{2}} = x \to 0 lim \frac{- 1/2 + O ( x ^{2} )}{- 1/6 + O ( x ^{2} )} = 3.

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