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Question
Evaluate the following limit :
`lim_(x -> pi) [(sqrt(1 - cosx) - sqrt(2))/(sin^2 x)]`
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Solution
`lim_(x -> pi) (sqrt(1 - cosx) - sqrt(2))/(sin^2 x)`
= `lim_(x -> pi) (sqrt(1 - cosx) - sqrt(2))/(sin^2x) xx (sqrt(1 - cosx) + sqrt(2))/(sqrt(1 - cosx) + sqrt(2))`
= `lim_(x -> pi) ((1 - cos x) - 2)/(( 1 - cos^2x)(sqrt(1 - cos x) + sqrt(2))`
= `lim_(x -> pi) (-(1 + cos x))/((1 + cos x)(1 - cosx)(sqrt(1 - cosx) + sqrt(2))`
= `lim_(x -> pi) (-1)/((1 - cosx)(sqrt(1 - cosx) + sqrt(2))) ...[(because x -> pi"," x ≠ pi),(therefore cos x ≠ cos pi = -1),(therefore 1 + cos x ≠ 0)]`
= `(lim_(x -> pi) (-1))/([lim_(x -> pi) (1 - cosx)] xx [lim_(x -> pi) (sqrt(1 - cosx) + sqrt(2))]`
= `(-1)/((1 - cos pi) (sqrt(1 - cos pi) + sqrt(2))`
= `(-1)/((1 + 1)(sqrt(1 + 1) + sqrt(2))`
= `(-1)/(4sqrt(2))`.
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