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 value of \( f(0) \) that makes the function \( f(x) = \frac{\sin x}{x} \) continuous at \( x = 0 \), we need to follow these steps:

### Step 1: Understand the condition for continuity
A function \( f(x) \) is continuous at a point \( x = a \) if:
\[ \lim_{x \to a} f(x) = f(a) \]
In this case, we need to check the continuity at \( x = 0 \).

### Step 2: Set up the limit
We need to find:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin x}{x} \]
Since \( f(x) \) is defined for \( x \neq 0 \), we will evaluate the limit as \( x \) approaches 0.

### Step 3: Evaluate the limit
Using the standard limit result:
\[ \lim_{h \to 0} \frac{\sin h}{h} = 1 \]
we can substitute \( h \) with \( x \):
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

### Step 4: Set the limit equal to \( f(0) \)
For \( f(x) \) to be continuous at \( x = 0 \), we need:
\[ f(0) = \lim_{x \to 0} f(x) = 1 \]

### Step 5: Conclusion
Thus, for \( f(x) \) to be continuous at \( x = 0 \), we must define:
\[ f(0) = 1 \]

### Final Answer
The function \( f(x) \) can be continuous at \( x = 0 \) if \( f(0) = 1 \).

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