Question

what exactly is going on in the highlighted region of this limit evaluated as x approaches pi/4?

http://i47.tinypic.com/2akkxvq.png

i understand how you eventually come up with those two parts of the limit and evaluate them separately, but why is the highlighted part -1?

Answer 1

\[\lim\limits_{x\rightarrow\pi/4}\frac1{\cos(x)}\cdot\boxed{\dfrac{\cos(x)-\sin(x)}{\sin(x)-\cos(x)}}\]\[=\lim\limits_{x\rightarrow\pi/4}\frac1{\cos(x)}\cdot\boxed{\dfrac{-1\left(\sin(x)-\cos(x)\right)}{\sin(x)-\cos(x)}}\]\[=\lim\limits_{x\rightarrow\pi/4}\frac1{\cos(x)}\cdot\boxed{-1}\]

Answer 2

the numerator has been re-arranged and the common factor of sin x-cos x cancels