 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/12/a/i/solution

sinh x = x + \frac{x ^{3}}{6} + O (x^{5}), cosh x = 1 + \frac{x ^{2}}{2} + O (x^{4}) .

Dividing these power series gives the hyperbolic tangent expansion

tanh x = (x + \frac{x ^{3}}{6} + O (x^{5})) (1 - \frac{x ^{2}}{2} + O (x^{4})) = x - \frac{x ^{3}}{3} + O (x^{5}) .

x \to 0

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