Find $\lim\limits_{x\to\pi/4} \frac{1-\tan(x)^2}{\sqrt{2}*\cos(x)-1}$ without using L'Hôpital's rule. Find $$\lim_{x\to\pi/4} \frac{1-\tan(x)^2}{\sqrt{2}\times \cos(x)-1}$$ without using L'Hôpital's rule.
I can solve it using L'Hôpital's rule, but is it possible to solve it without using L'Hôpital's rule?

Just another way to do it.
Let $x=y+\frac \pi 4$, expand and simplify. You should get $$\lim_{x\to\pi/4} \frac{1-\tan^2(x)}{\sqrt{2}\times \cos(x)-1}=\lim_{y\to 0}\frac{2 (\cos(y)-\sin (y)+1)}{(\cos (y)-\sin (y))^2}$$ which looks very simple.
