How do I to find the limits of these? $$\lim _{x\to \frac{\pi }{4}}\left(\frac{1-\sin 2x}{\cos x -\sin x}\right)$$
$$\lim _{x\to \frac{\pi}{4}}\left(\frac{\cos^5x - \sin^5x}{\cos\:2x}\right)$$
I tried to solve but I get 0/0.
Help me to find the limit.

For the first limit write the numerator as $\cos^2{x}+\sin^2{x}-2\sin{x}\cos{x}=\left(\cos{x}-\sin{x}\right)^2$ where we have used $\sin{2x}=2\sin{x}\cos{x}$ and $\cos^2{x}+\sin^2{x}$. The fraction simplify to
$$\cos{x}-\sin{x}\to 0$$
when $x\to\pi/4$
For the second limit we use
$$\cos{2x}=\cos^2{x}-\sin^2{x}=(\cos{x}-\sin{x})(\cos{x}+\sin{x})$$
and
$$a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$$
to simplify the fraction to
$${\cos^4{x}+\cos^3{x}\sin{x}+\cos^2{x}\sin^2{x}+\cos{x}\sin^3{x}+\sin^4{x}\over \cos{x}+\sin{x}}$$
and the limit of this fraction is
$${5\sqrt{2}\over 8}$$
