Question

$\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\cos x-\sin x}{\left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)\left(\cos x+\sin x\right)}$

Answer 1

$$\begin{gathered} \underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\cos x-\sin x}{\left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)\left(\cos x+\sin x\right)} \\ \text{Dividing the numerator and the denominator by}\sqrt{2}\text{:} \\ \underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{{\displaystyle \frac{1}{\sqrt{2}}}\cos x-{\displaystyle \frac{1}{\sqrt{2}}}\sin x}{\left(\frac{\mathrm{\pi }}{4}-x\right)\frac{\left(\cos x+\sin x\right)}{\sqrt{2}}} \\ =\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sqrt{2}\left(\sin {\displaystyle \frac{\mathrm{\pi }}{4}}\cos x-\cos {\displaystyle \frac{\mathrm{\pi }}{4}}\sin x\right)}{\left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)\left(\cos x+\sin x\right)} \\ =\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sqrt{2}\left(\sin \left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)\right)}{\left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)\left(\cos x+\sin x\right)} \\ =\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sqrt{2}}{\sin x+\cos x}\times \underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sin \left(\frac{\mathrm{\pi }}{4}-x\right)}{\left(\frac{\mathrm{\pi }}{4}-x\right)} \\ \Rightarrow \frac{\sqrt{2}}{{\displaystyle \frac{1}{\sqrt{2}}}+{\displaystyle \frac{1}{\sqrt{2}}}}\times 1 \\ =1 \end{gathered}$$