 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/2/a/solution

Put

h = x - π /2

tan (2 x) = tan (π + 2 h) = tan (2 h)

tan (2 h) = 2 h + O (h^{3})

therefore yields

x \to π /2 lim \frac{tan ( 2 x )}{x - π /2} = 2.

Although

tan x

is singular at

π /2

, the numerator here is

tan (2 x)

, which is smooth and zero there.

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