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