Question

Lim x-> pi/4 of (1-tanx)/(sinx-cosx)

Answer 1

`\lim_(x \to \ \pi /4) (1-tan x)/(sin x - cos x)`If we use: x=π/4 we will get:
`(1-1)/( √(2) /2- √(2)/2 ) = (0)/(0)` and thet is not defined limit. So we will transform the expression:
`(1- tan x)/(sin x - cos x)= (1- (sin x)/(cos x) )/(sin x - cos x)= ( (cos x - sin x)/(cos x) )/(sin x - cos x)= (-(sin x - cos x ))/(cos x( sin x - cos x) )= (-1)/(cos x)= (-1)/(cos ( \pi )/(4) )`Finally:
`(-1)/( ( √(2) )/(2) ) = (-2)/( √(2) ) =- √(2)` And that´s it. Don´t forget to put "lim" in every line, except when you substitute variable x with π/4. Thank you.