Question

$\underset{x\to \pi /4}{lim}\frac{\sqrt{2}\cos x-1}{\cot x-1}$ equals

• A. $1$
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\sqrt{2}$

Answer 1

Answer: (B)

$\frac{1}{2}$

$\underset{x\to \pi /4}{lim}\frac{\sqrt{2}\cos x-1}{\cot x-1}$

$=\underset{x\to \pi /4}{lim}\frac{\sqrt{2}\cos x-1}{\cot x-1}.\frac{\sqrt{2}\cos x+1}{\sqrt{2}\cos x+1}$

$=\underset{x\to \pi /4}{lim}\frac{2{\cos }^{2}x-1}{\cot x-1}.\frac{1}{\sqrt{2}\cos x+1}$

$=\underset{x\to \pi /4}{lim}\frac{({\cos }^{2}x-{\sin }^{2}x)\sin x}{\cos x-\sin x}.\frac{1}{\sqrt{2}\cos x+1}$

$=\underset{x\to \pi /4}{lim}(\cos x+\sin x)\sin x.\frac{1}{\sqrt{2}\cos x+1}=\frac{1}{2}$