Find the value of `lim_(x to 1) (1+sin pix)^(cos pix)`

Answer

To solve the limit \( \lim_{x \to 1} (1 + \sin(\pi x))^{\cos(\pi x)} \), we will follow these steps:

### Step 1: Substitute \( x = 1 \)
First, we substitute \( x \) with \( 1 \) in the expression:
\[ 1 + \sin(\pi \cdot 1) = 1 + \sin(\pi) = 1 + 0 = 1 \]
\[ \cos(\pi \cdot 1) = \cos(\pi) = -1 \]

### Step 2: Rewrite the limit
Now we can rewrite the limit:
\[ \lim_{x \to 1} (1 + \sin(\pi x))^{\cos(\pi x)} = \lim_{x \to 1} (1)^{-1} \]

### Step 3: Evaluate the limit
Since \( 1^{-1} = 1 \), we have:
\[ \lim_{x \to 1} (1 + \sin(\pi x))^{\cos(\pi x)} = 1 \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{1} \]