Question

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

Answer 1

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

Explanation:

`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`