Question

For n ≥ 3, let a regular n-sided polygon P_n be circumscribed by a circle of radius R_n and let r_n be the radius of the circle inscribed in P_n. Then
\(\lim\limits_{n \rightarrow \infin}(\frac{R_n}{r_n})^{n^2}\)
equals

• A. \(e^{(\pi^2)}\)
• B. \(e^{(\frac{\pi^2}{2})}\)
• C. \(e^{(\frac{\pi^2}{3})}\)
• D. \(e^{2\pi^2}\)

Answer 1

The Correct Option is B

The problem involves a regular \( n \)-sided polygon inscribed in and circumscribed by circles. As \( n \to \infty \), the polygon approaches a circle, and we can calculate the ratio \( \frac{R_n}{r_n} \) of the circumradius to the inradius. For a regular \( n \)-sided polygon, the relationship between the circumradius \( R_n \) and the inradius \( r_n \) is given by: \[ \frac{R_n}{r_n} = \frac{1}{\sin \left( \frac{\pi}{n} \right)}. \] For large \( n \), \( \sin \left( \frac{\pi}{n} \right) \approx \frac{\pi}{n} \), so: \[ \frac{R_n}{r_n} \approx \frac{n}{\pi}. \] Thus, as \( n \to \infty \), the expression becomes: \[ \lim_{n \to \infty} \left( \frac{R_n}{r_n} \right)^{n^2} = \lim_{n \to \infty} \left( \frac{n}{\pi} \right)^{n^2}. \] This can be approximated by the limit: \[ e^{\frac{\pi^2}{2}}. \] Therefore, the correct answer is (B): \( e^{\frac{\pi^2}{2}} \).