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Let x = π/4 + Δx So our limit is as Δx -> 0

tan π/4 = 1 sin 45 = 1/√2

numerator = 1 - tan (π/4 + Δx ) = 1 - [ (tan 45 + tan Δx)/(1 - tan 45 tan Δx) ]

= 1 - [ (1 + tan Δx)/ (1-tan Δx) ] = - 2 tan Δx / (1 - tan Δx)

Numerator / Δx = - 2 ( tan Δx / Δx ) * 1/(1 - tan Δx)

Lim Numerator / Δx = - 2 as Lim (tan Δx / Δx) = 1 and Lim tan Δx = 0

Δx->0 Δx->0 Δx->0

Denom = 1 - √2 sin (45+Δx) = 1 - Cos Δx - Sin Δx

= 2 sin² Δx/2 - 2 Sin Δx/2 Cos Δx/2 = 2 Sin Δx/2 [ sin Δx/2 - cos Δx/2]

Denom / Δx = ( Sin Δx/2 / Δx/2) [ sin Δx/2 - cos Δx/2 ]

Lim Denom / Δx = -1 as Lim sin Δx/2 / Δx/2 = 1

Δx -> 0 Δx/2 -> 0

Lim Δx/2 ->0 of Sin Δx/2 =0 and for cos , it is 1.

tan π/4 = 1 sin 45 = 1/√2

numerator = 1 - tan (π/4 + Δx ) = 1 - [ (tan 45 + tan Δx)/(1 - tan 45 tan Δx) ]

= 1 - [ (1 + tan Δx)/ (1-tan Δx) ] = - 2 tan Δx / (1 - tan Δx)

Numerator / Δx = - 2 ( tan Δx / Δx ) * 1/(1 - tan Δx)

Lim Numerator / Δx = - 2 as Lim (tan Δx / Δx) = 1 and Lim tan Δx = 0

Δx->0 Δx->0 Δx->0

Denom = 1 - √2 sin (45+Δx) = 1 - Cos Δx - Sin Δx

= 2 sin² Δx/2 - 2 Sin Δx/2 Cos Δx/2 = 2 Sin Δx/2 [ sin Δx/2 - cos Δx/2]

Denom / Δx = ( Sin Δx/2 / Δx/2) [ sin Δx/2 - cos Δx/2 ]

Lim Denom / Δx = -1 as Lim sin Δx/2 / Δx/2 = 1

Δx -> 0 Δx/2 -> 0

Lim Δx/2 ->0 of Sin Δx/2 =0 and for cos , it is 1.

**So numerator / Denominator as Lim Δx->0 , -2 / -1 = 2**