Evaluate the following limits :
`Lim_(x to pi) (sin x )/(x - pi)`

Answer

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

### Step 1: Rewrite the Limit
We start with the limit:
\[ \lim_{x \to \pi} \frac{\sin x}{x - \pi} \]
To simplify this, we can use the identity for sine, which states that \( \sin(\pi + \theta) = -\sin(\theta) \). Here, we can rewrite \( x \) as \( \pi + (x - \pi) \).

### Step 2: Substitute \( x \)
Substituting \( x = \pi + (x - \pi) \):
\[ \sin x = \sin(\pi + (x - \pi)) = -\sin(x - \pi) \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to \pi} \frac{-\sin(x - \pi)}{x - \pi} \]

### Step 3: Apply the Limit
Now, we can evaluate the limit:
\[ \lim_{x \to \pi} \frac{-\sin(x - \pi)}{x - \pi} \]
As \( x \) approaches \( \pi \), \( x - \pi \) approaches \( 0 \). We can use the standard limit:
\[ \lim_{y \to 0} \frac{\sin y}{y} = 1 \]
Let \( y = x - \pi \). Then as \( x \to \pi \), \( y \to 0 \):
\[ \lim_{y \to 0} \frac{-\sin y}{y} = -1 \]

### Step 4: Conclusion
Thus, we find that:
\[ \lim_{x \to \pi} \frac{\sin x}{x - \pi} = -1 \]

Therefore, the final answer is:
\[ \boxed{-1} \]

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