 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/19/c/i/solution

A trigonometric simplification avoids repeated differentiation:

\frac{t a n x - s i n x}{s i n ^{3} x} = \frac{1 - c o s x}{c o s x s i n ^{2} x} = \frac{1}{c o s x ( 1 + c o s x )}

for sufficiently small nonzero

x

. Continuity of cosine then gives

x \to 0 lim \frac{tan x - sin x}{sin ^{3} x} = \frac{1}{2} .

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