Question

If $f(x) = \frac{k \cos x}{\pi - 2x}$ for $x \neq \frac{\pi}{2}$ and $f\left(\frac{\pi}{2}\right) = 3$ is continuous at $x = \frac{\pi}{2}$, then the value of $k$ is:

• A. $6$
• B. $3$
• C. $2$
• D. $1.5$

Hint:

Apply L'Hopital's Rule for $\frac{0}{0}$ limits: differentiate numerator and denominator with respect to $x$: $\lim_{x \to \pi/2} \frac{-k\sin x}{-2} = \frac{k}{2} = 3 \implies k = 6$. It's super fast!

Answer 1

The Correct Option is A

Step 1: Concept
For a function $f(x)$ to be continuous at $x = a$, we must have $\lim_{x \to a} f(x) = f(a)$.

Step 2: Meaning
Here, we require $\lim_{x \to \pi/2} \frac{k \cos x}{\pi - 2x} = 3$.

Step 3: Analysis
Let $x = \frac{\pi}{2} + h$. As $x \to \frac{\pi}{2}$, $h \to 0$. \[ \lim_{h \to 0} \frac{k \cos\left(\frac{\pi}{2} + h\right)}{\pi - 2\left(\frac{\pi}{2} + h\right)} = \lim_{h \to 0} \frac{-k \sin h}{-2h} = \lim_{h \to 0} \frac{k \sin h}{2h} \] Since $\lim_{h \to 0} \frac{\sin h}{h} = 1$: \[ \frac{k}{2} = 3 \implies k = 6 \]

Step 4: Conclusion
The value of $k$ for the function to be continuous is $6$.

Final Answer: (A)