Q57 / 89 · KVPY 2017 (5 Nov SB/SX) · Mathematics
Let $\mathrm{x}_{\mathrm{n}}=\left(2^{\mathrm{n}}+3^{\mathrm{n}}\right)^{1 / 2 \mathrm{n}}$ for all natural numbers $\mathrm{n}$. Then
- A$\lim _{n \rightarrow \infty} x_{n}=\infty$
- Correct answerB$\lim _{n \rightarrow \infty} x_{n}=\sqrt{3}$
- C$\lim _{n \rightarrow \infty} x_{n}=\sqrt{3}+\sqrt{2}$
- D$\lim _{n \rightarrow \infty} x_{n}=\sqrt{5}$
Solution
$\lim _{n \rightarrow \infty}\left(3^{n}\right)^{1 / 2 n}\left(\left(\frac{2}{3}\right)^{n}+1\right)^{1 / 2 n}$
Put $\lim _{n \rightarrow \infty} \sqrt{3}$
Hence, the correct answer is option: 'B'.