[tex]3.\ \lim_{x \to \frac{\pi }{4}} \frac{1-2sin^2x}{sinx-cosx} \\\\= \lim_{x \to \frac{\pi }{4}} \frac{(1-sin^2x)-sin^2x}{sinx-cosx} \\\\= \lim_{x \to \frac{\pi }{4}} \frac{cos^2x-sin^2x}{sinx-cosx} \\\\= \lim_{x \to \frac{\pi }{4}} \frac{(cosx+sinx)(cosx-sinx)}{-(cosx-sinx)} \\\\= \lim_{x \to \frac{\pi }{4}} \frac{(cosx+sinx)}{-1} \\\\= \lim_{x \to \frac{\pi }{4}} (-cosx-sinx) \\\\= -cos(\frac{\pi}{4} )-sin(\frac{\pi}{4} ) \\\\=-\frac{1}{2} \sqrt{2}-\frac{1}{2} \sqrt{2}\\\\=-\sqrt{2}[/tex]
[tex]5.\ \lim_{x \to 0} \frac{1-cos^2x}{x.tan2x} \\\\= \lim_{x \to 0} \frac{sin^2x}{x.\frac{sin2x}{cos2x} } \\\\= \lim_{x \to 0} \frac{sin^2x.cos2x}{x.sin2x } \\\\= \lim_{x \to 0} \frac{sin^2x.cos2x}{x.2sinxcosx } \\\\= \lim_{x \to 0} \frac{sinx.cos2x}{2xcosx } \\\\= \lim_{x \to 0} \frac{cos2x}{2cosx } . \lim_{x \to 0} \frac{sinx}{x } \\\\= \frac{cos2(0)}{2cos(0) } . (1) \\\\= \frac{cos0}{2cos(0) } \\\\= \frac{1}{2(1) } \\\\= \frac{1}{2} \\[/tex]
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