Question

The value of \(\lim_{x \to \pi/2} \dfrac{\cos x}{x - \pi/2}\) is:

• A. 1
• B. -1
• C. 0
• D. \(\pi\)

Hint:

For limits of the form \(\frac{0}{0}\), always consider applying L'Hôpital's Rule or expansions like Taylor series near the point.

Answer 1

The Correct Option is B

Step 1: Check form of the limit.
As \(x \to \pi/2\), \(\cos x \to 0\) and denominator \(x - \pi/2 \to 0\). Thus, this is a \(\tfrac{0}{0}\) indeterminate form. Step 2: Apply L'Hôpital's Rule.
\[ \lim_{x \to \pi/2} \frac{\cos x}{x - \pi/2} = \lim_{x \to \pi/2} \frac{-\sin x}{1}. \] Step 3: Evaluate.
At \(x = \pi/2\): \[ -\sin\left(\frac{\pi}{2}\right) = -1. \] Final Answer: \[ \boxed{-1} \]