Evaluate the following limits :
`Lim_(x to 0 ) (sin x^(@))/x`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\sin x^\circ}{x} \), we can follow these steps:

### Step-by-Step Solution:

1. **Convert Degrees to Radians**:
We know that \( \sin x^\circ \) can be expressed in terms of radians. The conversion from degrees to radians is given by:
\[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \]
Therefore, we can rewrite \( \sin x^\circ \) as:
\[ \sin x^\circ = \sin\left(\frac{\pi}{180} \cdot x\right) \]

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

3. **Factor Out the Constant**:
We can factor out the constant \( \frac{\pi}{180} \) from the sine function:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi}{180} \cdot x\right)}{x} = \lim_{x \to 0} \frac{\sin\left(\frac{\pi}{180} \cdot x\right)}{\frac{\pi}{180} \cdot x} \cdot \frac{\pi}{180} \]

4. **Use the Standard Limit**:
We know that:
\[ \lim_{u \to 0} \frac{\sin u}{u} = 1 \]
In our case, let \( u = \frac{\pi}{180} \cdot x \). As \( x \to 0 \), \( u \to 0 \) as well. Thus, we can apply this limit:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi}{180} \cdot x\right)}{\frac{\pi}{180} \cdot x} = 1 \]

5. **Combine the Results**:
Now, substituting back, we have:
\[ \lim_{x \to 0} \frac{\sin x^\circ}{x} = 1 \cdot \frac{\pi}{180} = \frac{\pi}{180} \]

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