Question

Find the value of \(k\) if the function \(f(x) = \dfrac{k\sin x}{x}\) is continuous at \(x = 0\) and \(f(0)=3\).

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(6\)

Hint:

Whenever continuity at \(x=0\) involves expressions like \( \frac{\sin x}{x} \), remember the standard limit: \[ \lim_{x\to0}\frac{\sin x}{x}=1 \] This limit is very commonly used in calculus problems.

Answer 1

The Correct Option is C

Topic: Continuity and Trigonometric Limits
Step 1: Understanding the Question:
A function is continuous at a point if the limit of the function as \(x\) approaches that point is equal to the value of the function at that point.
Here, we need to ensure \(\lim_{x \to 0} f(x) = f(0)\).
Step 2: Key Formula or Approach:
We use the fundamental trigonometric limit:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Step 3: Detailed Explanation:
1. Set up the continuity equation:
\[ \lim_{x \to 0} \frac{k\sin x}{x} = f(0) \]
2. Apply the limit property:
\[ k \left( \lim_{x \to 0} \frac{\sin x}{x} \right) = f(0) \]
3. Substitute the known limit value and the given function value \(f(0) = 3\):
\[ k(1) = 3 \]
4. Solve for \(k\):
\[ k = 3 \]
Step 4: Final Answer:
The value of \(k\) that makes the function continuous is \(3\).