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Question: Answered & Verified by Expert
Let $\mathrm{x}_{\mathrm{n}}=\left(2^{\mathrm{n}}+3^{\mathrm{n}}\right)^{1 / 2 \mathrm{n}}$ for all natural numbers $\mathrm{n}$. Then
MathematicsLimitsKVPYKVPY 2017 (5 Nov SB/SX)
Options:
  • A $\lim _{n \rightarrow \infty} x_{n}=\infty$
  • B $\lim _{n \rightarrow \infty} x_{n}=\sqrt{3}$
  • C $\lim _{n \rightarrow \infty} x_{n}=\sqrt{3}+\sqrt{2}$
  • D $\lim _{n \rightarrow \infty} x_{n}=\sqrt{5}$
Solution:
1887 Upvotes Verified Answer
The correct answer is: $\lim _{n \rightarrow \infty} x_{n}=\sqrt{3}$
$\lim _{n \rightarrow \infty}\left(3^{n}\right)^{1 / 2 n}\left(\left(\frac{2}{3}\right)^{n}+1\right)^{1 / 2 n}$
Put $\lim _{n \rightarrow \infty} \sqrt{3}$

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