 Show that lim nr2 sin(2π/n) = πr2. I recommend making the change of variables similar to
 n→∞ 2 cos(2π/n)
what we did in class, letting k = 2π/n. If you do this, you must solve for n in terms of k, and determine what happens to k as n → ∞.

We want to show that:

\[
\lim_{n \to \infty} n r^2 \sin\left(\frac{2\pi}{n}\right) = \pi r^2
\]

### Step 1: Change of Variables
Let’s make the suggested change of variables \( k = \frac{2\pi}{n} \), which implies that \( n = \frac{2\pi}{k} \).

Now, as \( n \to \infty \), we have \( k \to 0 \) because \( k = \frac{2\pi}{n} \).

Thus, the original limit becomes:

\[
\lim_{k \to 0} \frac{2\pi}{k} r^2 \sin(k)
\]

### Step 2: Use of the Small Angle Approximation
For small \( k \), we can use the approximation:

\[
\sin(k) \approx k \quad \text{as} \quad k \to 0
\]

Therefore, the expression becomes:

\[
\lim_{k \to 0} \frac{2\pi}{k} r^2 k = \lim_{k \to 0} 2\pi r^2 = \pi r^2
\]

### Conclusion:
Thus, we have shown that:

\[
\lim_{n \to \infty} n r^2 \sin\left(\frac{2\pi}{n}\right) = \pi r^2
\]

Would you like more details on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. How does the small angle approximation for \(\sin(x)\) work, and why is it valid as \(x \to 0\)?
2. What would happen if we used a different substitution for \( n \to \infty \)?
3. Can you explain why \( n r^2 \sin\left(\frac{2\pi}{n}\right) \) approaches the area of a circle as \( n \to \infty \)?
4. How can this limit be related to calculating the area of a regular polygon inscribed in a circle?
5. What other trigonometric limits can benefit from small-angle approximations?

**Tip:** When working with limits, especially involving trigonometric functions, always consider small-angle approximations, as they often simplify the process.

Show that lim nr2 sin(2π/n) = πr2. I recommend making the change of variables similar to n→∞ 2 cos(2π/n) what we did in class, letting k = 2π/n. If you do this, you must solve for n in terms of k, and determine what happens to k as n → ∞.

Explore the detailed step-by-step solution to the limit lim(n→∞) nr² sin(2π/n) = πr², incorporating a change of variables and small angle approximation. Understand key concepts like limits and trigonometric approximations, suitable for undergraduate mathematics students.

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