 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/19/b/solution

Factor out the dominant cubic term and use a geometric series in

1/ (3 x^{2})

\frac{1}{x + 3 x ^{3}} = \frac{1}{3 x ^{3}} (1 + \frac{1}{3 x ^{2}})^{- 1} = \frac{1}{3 x ^{3}} - \frac{1}{9 x ^{5}} + O (x^{- 7}) .

Thus the requested leading term and absolute error order are

g (x) = \frac{1}{3 x ^{3}} + O (x^{- 5}) (x \to + \infty) .

The relative error of retaining only the leading term is

O (x^{- 2})

; it differs from the absolute error order

O (x^{- 5})

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