 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/ii/solution

The linear terms cancel:

tan x = x + x^{3} /3 + O (x^{5})

, whereas

tanh x = x - x^{3} /3 + O (x^{5})

. Also

sinh x - x = x^{3} /6 + O (x^{5})

. Their leading nonzero terms therefore give

x \to 0 lim \frac{tan x - tanh x}{sinh x - x} = \frac{2/3}{1/6} = 4.

The Maclaurin series remainder orders justify division after factoring out

x^{3}

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