Let `f(x)=(sin x)/(x), x ne 0`. Then f(x) can be continous at x=0, if

Options

A. `f(0)=0`
B. `f(0)=1`
C. `f(0)=2`
D. `f(0)=-2`

Explanation

To determine the continuity of the function \( f(x) = \frac{\sin x}{x} \) at \( x = 0 \), we need to follow these steps:

### Step 1: Define Continuity at a Point
A function \( f(x) \) is continuous at a point \( c \) if:
1. \( f(c) \) is defined.
2. \( \lim_{x \to c} f(x) \) exists.
3. \( \lim_{x \to c} f(x) = f(c) \).

### Step 2: Check if \( f(0) \) is Defined
Since the function \( f(x) \) is defined as \( \frac{\sin x}{x} \) for \( x \neq 0 \), we need to define \( f(0) \) in order to check continuity at that point.

### Step 3: Calculate the Limit as \( x \) Approaches 0
We need to find \( \lim_{x \to 0} f(x) \):
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin x}{x} \]
Using the standard limit result:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

### Step 4: Define \( f(0) \)
To make \( f(x) \) continuous at \( x = 0 \), we define:
\[ f(0) = 1 \]

### Step 5: Verify Continuity
Now we check the three conditions for continuity:
1. \( f(0) = 1 \) is defined.
2. \( \lim_{x \to 0} f(x) = 1 \) exists.
3. \( \lim_{x \to 0} f(x) = f(0) \).

Since all conditions are satisfied, \( f(x) \) can be continuous at \( x = 0 \) if we define \( f(0) = 1 \).

### Final Conclusion
Thus, \( f(x) \) can be continuous at \( x = 0 \) if \( f(0) = 1 \).

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