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We are given that \( f(x) = \sin x \) and need to evaluate the limit:

\[
\lim_{x \to 2\pi} \frac{f(2\pi) - f(x)}{x - 2\pi}
\]

This is essentially the definition of the derivative of \( f(x) \) at \( x = 2\pi \). Let's analyze this step by step.

### Step 1: Definition of the Derivative
The derivative of a function \( f(x) \) at a point \( x = a \) is given by:

\[
f'(a) = \lim_{x \to a} \frac{f(a) - f(x)}{a - x}
\]

In this case, \( a = 2\pi \) and \( f(x) = \sin x \). Therefore, the expression becomes:

\[
\lim_{x \to 2\pi} \frac{\sin(2\pi) - \sin(x)}{x - 2\pi}
\]

### Step 2: Evaluate \( \sin(2\pi) \)
We know that:

\[
\sin(2\pi) = 0
\]

Thus, the expression simplifies to:

\[
\lim_{x \to 2\pi} \frac{0 - \sin(x)}{x - 2\pi} = \lim_{x \to 2\pi} \frac{-\sin(x)}{x - 2\pi}
\]

### Step 3: Apply the Derivative of \( \sin(x) \)
We recognize that this is the derivative of \( \sin(x) \) at \( x = 2\pi \). The derivative of \( \sin(x) \) is:

\[
\frac{d}{dx} \sin(x) = \cos(x)
\]

Therefore:

\[
\lim_{x \to 2\pi} \frac{-\sin(x)}{x - 2\pi} = -\cos(2\pi)
\]

### Step 4: Evaluate \( \cos(2\pi) \)
We know that:

\[
\cos(2\pi) = 1
\]

Thus:

\[
-\cos(2\pi) = -1
\]

### Final Answer:
The value of the limit is \( -1 \), so the correct option is **B**.

---

Would you like a more detailed explanation of any step? 

Here are 5 related questions to consider:
1. What is the geometric interpretation of a derivative?
2. How do you find the derivative of \( \cos(x) \)?
3. What are common trigonometric limits and their applications?
4. How does L'Hopital's Rule work for evaluating limits involving 0/0 forms?
5. What is the second derivative of \( \sin(x) \), and how is it interpreted?

**Tip:** When evaluating trigonometric limits, recognizing derivative patterns can simplify the process greatly.

If f(x) = sin x, then lim (x→2π) [f(2π) - f(x)] / (x - 2π) = ?

Solve the limit problem lim (x→2π) [f(2π) - f(x)] / (x - 2π) where f(x) = sin x. This question covers trigonometric limits and the derivative of a sine function, perfect for students in Grades 11-12. Learn the step-by-step solution to find that the correct answer is -1.

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