Question Number 280 by arnav last updated on 25/Jan/15
$$\mathrm{Evaluate}\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\pi/\mathrm{4}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)} \\ $$
Answered by 123456 last updated on 18/Dec/14
$$\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)}\rightarrow\frac{\mathrm{0}}{\mathrm{0}} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{−\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}{−\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)+\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(−\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)} \\ $$$$=\frac{−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}{−\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)+\left(\frac{\pi}{\mathrm{4}}−\frac{\pi}{\mathrm{4}}\right)\left(−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)} \\ $$$$=\mathrm{1} \\ $$
Answered by 123456 last updated on 22/Dec/14
$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+{x}\right)}\centerdot\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}}} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)}{\frac{\pi}{\mathrm{4}}−{x}}\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+{x}\right)} \\ $$$$=\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{y}}{{y}}\centerdot\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{\pi}{\mathrm{4}}\right)} \\ $$$$=\mathrm{1}\centerdot\frac{\mathrm{1}}{\mathrm{sin}\:\frac{\pi}{\mathrm{2}}}=\mathrm{1} \\ $$