If `f (x) = sin (x)/(x), x ne 0` then the value of the function at x = 0 so that the function is continuous at x= 0 , is :

Options

A. 1
B. 0
C. -1
D. None of these

Answer

To determine the value of the function \( f(x) = \frac{\sin x}{x} \) at \( x = 0 \) so that the function is continuous at that point, we follow these steps:

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

### Step 2: Find the limit as \( x \) approaches 0
We need to calculate:
\[ \lim_{x \to 0} \frac{\sin x}{x} \]
This is a standard limit in calculus. We know from calculus that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

### Step 3: Assign the value to \( f(0) \)
To make the function continuous at \( x = 0 \), we set:
\[ f(0) = 1 \]

### Conclusion
Thus, the value of the function at \( x = 0 \) so that the function is continuous at \( x = 0 \) is:
\[ \boxed{1} \]

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