Question

Let \( f(x) = \sin x - \cos x \). Then the value of 

\[ \lim_{x \to \frac{\pi}{2}} \frac{f(x) - f\left(\frac{\pi}{2}\right)}{x - \frac{\pi}{2}} \]

is ______.

• A. \(0\)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{\sqrt{2}} \)
• D. \(1\)
• E. \( \sqrt{2} \)

Hint:

Limits of this form directly give derivative at the point.

Answer 1

The Correct Option is D

Concept: The given limit represents the derivative of the function at a point: \[ \lim_{x \to a} \frac{f(x)-f(a)}{x-a} = f'(a) \]

Step 1: Recognize the derivative form. \[ \lim_{x \to \frac{\pi}{2}} \frac{f(x)-f\left(\frac{\pi}{2}\right)}{x-\frac{\pi}{2}} = f'\left(\frac{\pi}{2}\right) \]

Step 2: Differentiate the function. \[ f(x) = \sin x - \cos x \] \[ f'(x) = \cos x + \sin x \]

Step 3: Evaluate at \(x = \frac{\pi}{2}\). \[ f'\left(\frac{\pi}{2}\right) = \cos\frac{\pi}{2} + \sin\frac{\pi}{2} \] \[ = 0 + 1 = 1 \]