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.