Question

$\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2}\cos x - 1}{\cot x - 1}$ is equal to:

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

Answer 1

The Correct Option is C

We begin by substituting $x = \frac{\pi}{4}$: \[ \lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2}\cos x - 1}{\cot x - 1}. \] For small values around $x = \frac{\pi}{4}$, use the approximations: \[ \cos x \approx 1 - \frac{(x - \frac{\pi}{4})^2}{2} \quad \text{and} \quad \cot x \approx 1 + (x - \frac{\pi}{4}). \] Substituting these approximations and simplifying, we find the limit to be $\frac{1}{2}$.