<s>[INST] 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?
 [/INST] You can try the following:
\begin{align}
\frac{1-\tan^2({x})}{\sqrt{2} \cos{x}-1}\frac{\sqrt{2} \cos{x}+1}{ \sqrt{2} \cos{x}+1 }&=\frac{1-\frac{\sin^2{x}}{\cos^2{x}}}{2 \cos^2{x}-1} (\sqrt{2} \cos{x}+1)\\ &=\frac{\cos^2{x}-\sin^2{x}}{\cos^2{x} (2\cos^2{x}-\sin^2{x}-\cos^2{x}) }(\sqrt{2} \cos{x}+1)\\
&=\frac{\sqrt{2} \cos{x}+1}{\cos^2x}.
\end{align}
Now take the limit as usual.
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