Question

Evaluate
lim x->pi/4 ?2 cosx-1/cot x - 1

Evaluate: $\underset{\text{x → π/4}}{\lim }\frac{\sqrt{2}\text{cos x - 1}}{\text{cot x - 1}}$

Answer 1

$$\begin{gathered} \underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sqrt{2}\cos \mathrm{x}-1}{\cot \mathrm{x}-1} \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{{\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}\left[\sqrt{2}\cos \mathrm{x}-1\right]}{{\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}\left(\cot \mathrm{x}-1\right)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[\mathbf{Using}\mathbf{}\mathbf{L}\mathbf{\text{'}}\mathbf{Hospital}\mathbf{}\mathbf{rule}\right] \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\left[-\sqrt{2}\sin \mathrm{x}\right]}{-{\mathrm{cosec}}^2\mathrm{x}} \\ =\sqrt{2}\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sin \mathrm{x}}{{\mathrm{cosec}}^2\mathrm{x}} \\ =\sqrt{2}\times \underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }{\sin }^3\mathrm{x} \\ =\sqrt{2}\times {\left(\frac{1}{\sqrt{2}}\right)}^3 \\ =\sqrt{2}\times \frac{1}{2\sqrt{2}} \\ =\frac{1}{2} \end{gathered}$$