Find $\mathop {\lim }\limits_{x \to \frac{1}{2}} \left( {\frac{{\left( {\frac{1}{4} - {x^2}} \right)}}{{\cot \pi x}}} \right)$.

using l’Hôpital’s Rule     (M1)

$\mathop {\lim }\limits_{x \to \frac{1}{2}} \left( {\frac{{\left( {\frac{1}{4} - {x^2}} \right)}}{{\cot \pi x}}} \right) = \mathop {\lim }\limits_{x \to \frac{1}{2}} \left[ {\frac{{ - 2}}{{ - \pi {\text{cose}}{{\text{c}}^2}\pi x}}} \right]$     A1A1

$= \frac{{ - 1}}{{ - \pi {\text{cose}}{{\text{c}}^2}\frac{\pi }{2}}} = \frac{1}{\pi }$     (M1)A1

[5 marks]

This question was accessible to the vast majority of candidates, who recognised that L’Hôpital’s rule was required. However, some candidates omitted the factor $\pi$ in the differentiation of ${\cot \pi x}$. Some candidates replaced ${\cot \pi x}$ by $\cos \pi x{\text{/}}\sin \pi x$, which is a valid method but the extra algebra involved often led to an incorrect answer. Many fully correct solutions were seen.