 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ia/paper-1/9d/iv/solution

For the final limit, near zero

cos x > 0

, so the real logarithm is well defined. First apply the proved rule to

tan x / x

on the right: its derivative ratio is

sec^{2} x \to 1

. Next apply it to

lo g (cos x) / x^{2}

\frac{- t a n x}{2 x} ⟶ - \frac{1}{2} .

The original quotient is even, so the left limit equals the right limit. Consequently

x \to 0 lim \frac{lo g cos x}{x ^{2}} = - \frac{1}{2} .

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