Introduction:
In this problem, we are asked to find the limit of the function as x approaches 1. The given function is (1 - sinx/2) / (π - x)^2. We will use algebraic manipulation and basic limit rules to evaluate the limit.

Step 1: Simplify the function:
Let's start by simplifying the given function. We can rewrite sinx/2 as (1/2) * sinx. Therefore, the function becomes (1 - (1/2) * sinx) / (π - x)^2.

Step 2: Factor out the common term:
Next, we can factor out (1/2) from the numerator, which gives us (1/2) * (2 - sinx) / (π - x)^2.

Step 3: Apply the limit rules:
To evaluate the limit, we can apply the limit rules. As x approaches 1, we can substitute the value of x into the function. Let's do that:

(1/2) * (2 - sin(1)) / (π - 1)^2.

Step 4: Evaluate the limit:
Now, we can evaluate the limit by substituting the value of sin(1) and simplifying further. Since sin(1) is a transcendental number, we cannot simplify it further. Therefore, the limit cannot be evaluated analytically.

Conclusion:
In conclusion, the limit of the given function as x approaches 1 cannot be determined analytically. However, we can approximate the limit using numerical methods or calculators.