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Question
Evaluate: `lim_(x -> 0) (sqrt(2) - sqrt(1 + cos x))/(sin^2x)`
Solution
Given that `lim_(x -> 0) (sqrt(2) - sqrt(1 + cos x))/(sin^2x)`
= `lim_(x -> 0) (sqrt(2) - sqrt(1 + cos x))/(sin^2x) xx (sqrt(2) + sqrt(1 + cosx))/(sqrt(2) + sqrt(1 + cos x))`
= `lim_(x -> 0) (2 - (1 + cos x))/(sin^2x [sqrt(2) + sqrt(1 + cos x)])`
= `lim_(x -> 0) (1 - cos x)/(sin^2 xx [sqrt(2) + sqrt(1 + cos x)])`
= `lim_(x -> 0) (2 sin^2 x/2)/((2 sin x/2 cos x/2)^2) xx 1/([sqrt(2) + sqrt(1 + cos x)])`
= `lim_(x -> 0) (2 sin^2 x/2)/(4 sin^2 x/2 cos^2 x/2) xx 1/([sqrt(2) + sqrt(1 + cos x)])`
= `lim_(x -> 0) 2/(4 cos^2 x/2) xx 1/([sqrt(2) + sqrt(1 + cos x)])`
Taking limit, we get
= `2/(4 cos^2 0) xx 1/((sqrt(2) + sqrt(2))`
= `1/2 xx 1/(2sqrt(2))`
= `1/(4sqrt(2))`
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