Question

$\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sqrt{2}-\cos x-\sin x}{{\left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)}^2}$

Answer 1

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