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How can I prove the following limit equals two without LHopital?

Through L'Hôpital, I reached this conclusion:

$$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\cos 2 x}{1-\sqrt{2} \sin x}=2$$

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HINT

\begin{align*} \cos(2x) = \cos^{2}(x) - \sin^{2}(x) = 1 - 2\sin^{2}(x) = (1 - \sqrt{2}\sin(x))(1+\sqrt{2}\sin(x)) \end{align*}

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  • $\begingroup$ Thank you so much $\endgroup$ – sec noor Dec 5 '20 at 0:53
  • $\begingroup$ You are welcome! I am glad to be of help. $\endgroup$ – APCorreia Dec 5 '20 at 1:29

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