Question

Find the value of \(k\) if the function \(f(x)=\dfrac{k\sin x}{x}\) for \(x\neq0\) and \(f(0)=3\) is continuous at \(x=0\).

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(4\)

Hint:

For continuity problems involving piecewise functions, the core task is to equate the limit of the function as it approaches the point with the defined value at that point. 
Recognizing standard limits like \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) is crucial for solving these questions quickly. 
 

 

Answer 1

The Correct Option is C

Step 1: Understanding the Question: 
We are given a piecewise function \(f(x)\) which is defined differently at \(x=0\) and for all other values of \(x\). We are told the function is continuous at \(x=0\) and we need to find the value of the constant \(k\) that makes this true. 
Step 2: Key Formula or Approach: 
For a function \(f(x)\) to be continuous at a point \(x=a\), the following condition must be met: 
\[ \lim_{x \to a} f(x) = f(a) \] In this problem, \(a=0\). We will also need the fundamental trigonometric limit: 
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \] Step 3: Detailed Explanation: 
(i) Apply the condition for continuity at x=0: 
For \(f(x)\) to be continuous at \(x=0\), we must have: 
\[ \lim_{x \to 0} f(x) = f(0) \] We are given that \(f(0) = 3\). So, the condition becomes: 
\[ \lim_{x \to 0} f(x) = 3 \] (ii) Calculate the limit of f(x) as x approaches 0: 
For the limit, we use the definition of \(f(x)\) when \(x \neq 0\): 
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{k\sin x}{x} \] We can take the constant \(k\) outside the limit: 
\[ = k \left( \lim_{x \to 0} \frac{\sin x}{x} \right) \] Using the standard limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), we get: 
\[ = k \times 1 = k \] (iii) Equate the limit and the function value: 
From step (i) and (ii), we equate the two results: 
\[ \lim_{x \to 0} f(x) = k \] and \[ f(0) = 3 \] Therefore, for continuity: 
\[ k = 3 \] Step 4: Final Answer: 
The value of \(k\) is 3.