How do you evaluate $(1-tanx) / (sinx - cos x)$ as x approaches pi/4?

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Answer 1 · 2016-04-14T19:18:37

$lim_(x->pi/4) (1-tanx)/(sinx-cosx)=-sqrt2$

$lim_(x->pi/4) (1-tanx)/(sinx-cosx)=(1-tan(pi/4))/(sin(pi/4)-cos(pi/4))=(1-1)/(1/sqrt2 -1/sqrt2)=0/0$

So this is one of the indeterminate type and we can use L'Hopital's rule $lim_(x->a) (f'(x))/(g'(x))$

$lim_(x->pi/4) (-sec^2x)/(cosx-(-sinx))=(-1/cos^2x)/(cosx+sinx)$

$=(-1/(1/sqrt2)^2)/(1/sqrt2 +1/sqrt2) = -2/(2/sqrt2) = -2xxsqrt2/2 =-sqrt2$

How do you evaluate (1-tanx) / (sinx - cos x) (1-tanx) / (sinx - cos x) as x approaches pi/4?