If the function `f(x)={{:(sinx,xne0),(k,x=0):}`
is continuous at x = 0 than find the value of k?

Options

A. 2
B. 1
C. `-1`
D. 0

Answer

To determine the value of \( k \) such that the function

\[ f(x) = \begin{cases} \sin x & \text{if } x \neq 0 \\ k & \text{if } x = 0 \end{cases} \]

is continuous at \( x = 0 \), we need to ensure that the left-hand limit (LHL) and the right-hand limit (RHL) at \( x = 0 \) are equal to the value of the function at that point, \( f(0) \).

### Step-by-Step Solution:

1. **Find the limit of \( f(x) \) as \( x \) approaches 0:**
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \sin x \]
The limit of \( \sin x \) as \( x \) approaches 0 is:
\[ \lim_{x \to 0} \sin x = 0 \]

2. **Set the limit equal to \( f(0) \):**
For the function to be continuous at \( x = 0 \), we need:
\[ \lim_{x \to 0} f(x) = f(0) \]
Therefore, we have:
\[ 0 = k \]

3. **Conclusion:**
The value of \( k \) that makes the function continuous at \( x = 0 \) is:
\[ k = 0 \]

### Final Answer:
The value of \( k \) is \( 0 \).

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