Putting z=x-pi/4 show that limit of (1-tanx)/(1-sqrt2 sinx)=2 as x approaches to pi/4?

1 Answer
Bio
Dec 12, 2017

#x = z+pi/4#

#tanx = tan(z+pi/4)#

#= frac{tanz+tan frac{pi}{4}}{1-tanz tan frac{pi}{4}}#

#= frac{tanz + 1}{1 - tanz (1)}#

#= frac{1+ tanz}{1 - tanz}#

#sinx = sin(z+pi/4)#

#= sinz cos frac{pi}{4} + cosz sin frac{pi}{4}#

#= 1/sqrt2 (sinz + cosz)#

#lim_{x->pi/4} (1 - tanx)/(1-sqrt2 sinx)#

#= lim_{z->0} (1 - (1+ tanz)/(1 - tanz))/(1-sqrt2 1/sqrt2 (sinz + cosz))#

#= lim_{z->0} ((1 -tan z - (1+ tanz))/(1 - tanz))/(1- sinz - cosz)#

#= lim_{z->0} (-2tanz)/((1- sinz - cosz)(1 - tanz))#

(Assuming limit exists)

#= lim_{z->0} (-2tanz)/(1- sinz - cosz)#

#= 2 lim_{z->0} (tanz)/(sinz + cosz - 1)#

#= 2 lim_{z->0} ((sinz + cosz - 1)/tanz)^(-1)#

#= 2 lim_{z->0} (sinz/tanz - (1-cosz)/tanz)^(-1)#

#= 2 lim_{z->0} (cosz - (1-cos^2z)/(tanz(1+cosz)))^(-1)#

#= 2 lim_{z->0} (1 - (sin^2z)/(tanz(1+cosz)))^(-1)#

#= 2 lim_{z->0} (1 - (sinzcosz)/(1+cosz))^(-1)#

#= 2 lim_{z->0} (1 - ((0)(1))/(1+1))^(-1)#

#= 2#

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