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Question
Evaluate the following :
`lim_(x -> pi/4) [(sinx - cosx)^2/(sqrt(2) - sinx - cosx)]`
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Solution
`lim_(x -> pi/4) [(sinx - cosx)^2/(sqrt(2) - sinx - cosx)]`
= `lim_(x -> pi/4) (1 - sin2x)/(sqrt(2) - (sin x + cos x))`
= `lim_(x -> pi/4) (1 - sin2x)/(sqrt(2) - sqrt(1 + sin2))`
Put 1 + sin 2x = t
∴ sin 2x = t – 1
As `x -> pi/4, "t" -> 1 + sin 2 (pi/4)`
∴ `"t" -> 1 + sin pi/2`
∴ t → 1 + 1
∴ t → 2
∴ Required limit
= `lim_("t" -> 2) (1 - ("t" - 1))/(sqrt(2) - sqrt("t"))`
= `lim_("t" -> 2) (2 - "t")/(2^(1/2) - "t"^(1/2))`
= `lim_("t" -> 2) ("t" - 2)/("t"^(1/2) - 2^(1/2))`
= `lim_("t" -> 2) 1/(("t"^(1/2) - 2^(1/2))/("t" - 2)) ...[("Divide Numerator and Denominator"),("by" "t" - 2 "As" "t" -> 2 "," "t" ≠ 2),(therefore "t" - 2 ≠ 0)]`
= `1/(1/2(2)^((-1)/2)`
= `2(2)^(1/2)`
= `2sqrt(2)`
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