Question

lim x->0 1+sinx-cosx/1-sinx-cosx

Answer 1

$frac{1+sin x - cos x}{1-sin x -cos x}=frac{(1+sin x - cos x)(1+sin x +cos x)}{(1-sin x -cos x)(1+sin x +cos x)}=\ frac{1+sin x-cos x+sin x+sin^2 x - sin x cos x+cos x + sin x cos x - cos^2 x}{1-sin x - cos x + sin x - sin^2 x - sin x cos x + cos x - sin x cos x - cos^2 x}=\ =frac{2sin x + 2sin^2 x}{-2sin x cos x}=frac{1+sin x}{-cos x}\ \ lim_{x to 0}frac{1+sin x - cos x}{1-sin x -cos x} = lim_{x to 0}frac{1+sin x}{-cos x}=-1$