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Question
`lim_(x -> pi) (1 - sin x/2)/(cos x/2 (cos x/4 - sin x/4))`
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Solution
Given, `lim_(x -> pi) (1 - sin x/2)/(cos x/2 (cos x/4 - sin x/4))`
= `lim_(x -> pi) (cos^2 x/4 + sin^2 x/4 - 2 sin x/4 * cos x/4)/((cos^2 x/4 - sin^2 x/4)(cos x/4 - sin x/4))` ......`[because cos 2theta = cos^2theta - sin^2theta]`
= `lim_(x -> pi) (cos x/4 - sin x/4)^2/((cos x/4 - sin x/4) * (cos x/4 + sin x/4) * (cos x/4 - sin x/4))`
= `lim_(x -> pi) 1/((cos x/4 + sin x x / 4))`
Taking limits we have
= `1/(cos pi/4 + sin pi/4)`
= `1/(1/sqrt(2) + 1/sqrt(2))`
= `(1/2)/(2/sqrt(2))`
= `1/sqrt(2)`
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