`lim_(x to 0) (sin x^(@))/( x)` is equal to

Options

A. 1
B. `pi`
C. `-pi`
D. `(pi)/(180)`

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin x^{(\circ)}}{x} \), we need to convert the angle from degrees to radians first.

### Step-by-Step Solution:

1. **Convert Degrees to Radians**:
We know that \( 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \). Therefore, \( x^{(\circ)} = x \cdot \frac{\pi}{180} \).

2. **Substituting into the Limit**:
Substitute \( x^{(\circ)} \) in the limit:
\[ \lim_{x \to 0} \frac{\sin(x^{(\circ)})}{x} = \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{x} \]

3. **Rewrite the Limit**:
To apply the standard limit \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \), we need to express the limit in a suitable form:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{x} = \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \cdot \frac{\frac{\pi x}{180}}{x} \]

4. **Simplifying the Expression**:
The expression can be simplified:
\[ = \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \cdot \frac{\pi}{180} \]

5. **Evaluate the Limit**:
As \( x \to 0 \), \( \frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \) approaches 1. Therefore:
\[ = 1 \cdot \frac{\pi}{180} = \frac{\pi}{180} \]

### Final Answer:
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin x^{(\circ)}}{x} = \frac{\pi}{180} \]