2014-07-31T16:29:05+05:30

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2014-07-31T18:53:14+05:30

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

Thanks a lot... I needed this badly :)