Evaluate the following limits :
`lim _( xto pi) ( sin x)/(pi - x)`

Answer

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

### Step 1: Rewrite the limit
We start with the limit:
\[ \lim_{x \to \pi} \frac{\sin x}{\pi - x} \]

### Step 2: Substitute \( \theta = \pi - x \)
To simplify the expression, we can make the substitution \( \theta = \pi - x \). Consequently, as \( x \) approaches \( \pi \), \( \theta \) approaches \( 0 \). We can express \( x \) in terms of \( \theta \):
\[ x = \pi - \theta \]
Thus, the limit becomes:
\[ \lim_{\theta \to 0} \frac{\sin(\pi - \theta)}{\pi - (\pi - \theta)} = \lim_{\theta \to 0} \frac{\sin(\pi - \theta)}{\theta} \]

### Step 3: Use the sine identity
Using the sine identity, we know that:
\[ \sin(\pi - \theta) = \sin \theta \]
So, we can rewrite the limit as:
\[ \lim_{\theta \to 0} \frac{\sin \theta}{\theta} \]

### Step 4: Evaluate the limit
It is a known limit that:
\[ \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \]
Thus, we conclude:
\[ \lim_{x \to \pi} \frac{\sin x}{\pi - x} = 1 \]

### Final Answer
The limit evaluates to:
\[ \boxed{1} \]