Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 20 / c

The cosine-difference identity gives

cos x - cos 3 x = 2 sin 2 x sin x

. Hence

\frac{c o s x - c o s 3 x}{x ^{2}} = 4 \frac{s i n 2 x}{2 x} \frac{s i n x}{x} .

Each sine ratio tends to one, so

x \to 0 lim \frac{cos x - cos 3 x}{x ^{2}} = 4.

Equivalently, the quadratic terms of the two cosine expansions differ by

4 x^{2}

ii

Near zero,

cos x = 1 - x^{2} /2 + O (x^{4})

and

ln (1 + v) = v + O (v^{2})

, so

ln cos x = - x^{2} /2 + O (x^{4})

. Replacing

x

3 x

similarly gives

ln cos 3 x = - 9 x^{2} /2 + O (x^{4})

. Therefore

x \to 0 lim \frac{ln cos x}{ln cos 3 x} = \frac{1}{9} .

For all sufficiently small nonzero

x

, both cosines are positive and the denominator logarithm is nonzero, so this quotient and its two-sided real limit are well-defined.

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