 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/20/b/iii/solution

To retain the finite term after cancellation, expand far enough before dividing. The Maclaurin series are

csc x = x^{- 1} + \frac{x}{6} + \frac{7 x ^{3}}{360} + O (x^{5}), sinh x = x + \frac{x ^{3}}{6} + \frac{x ^{5}}{120} + O (x^{7}) .

Thus

\frac{c s c x}{x ^{3}} - \frac{s i n h x}{x ^{5}} = (x^{- 4} + \frac{x ^{- 2}}{6} + \frac{7}{360}) - (x^{- 4} + \frac{x ^{- 2}}{6} + \frac{1}{120}) + O (x^{2}),

and hence

x \to 0 lim (\frac{csc x}{x ^{3}} - \frac{sinh x}{x ^{5}}) = \frac{1}{90} .

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