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Question
Evaluate the following limit :
`lim_(x -> pi/4) [(cosx - sinx)/(cos2x)]`
Solution
`lim_(x -> pi/4) (cosx - sinx)/(cos2x)`
= `lim_(x -> pi/4) (cosx - sinx)/(cos^2x - sin^2x)`
= `lim_(x -> pi/4) (cosx - sinx)/((cosx - sinx)(cosx + sinx))`
= `lim_(x -> pi/4) 1/(cosx + sinx) ...[(because x -> pi/4"," x ≠ pi/4),(therefore cos x ≠sinx),(therefore x - sin x ≠0)]`
= `(lim_(x -> pi/4) (1))/(lim_(x -> pi/4) (cosx + sinx))`
= `1/(cos(pi/4) + sin(pi/4))`
= `1/(1/sqrt(2) + 1/sqrt(2))`
= `1/((2/sqrt(2))`
= `1/sqrt(2)`.
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