AP EAMCET · Maths · Definite Integration
\(\lim _{x \rightarrow \infty}\left[\left(1+\frac{1}{n^3}\right)^{\frac{1}{n^3}}\left(1+\frac{8}{n^3}\right)^{\frac{4}{n^3}}\left(1+\frac{27}{n^3}\right)^{\frac{9}{n^3}} \ldots . .(2)^{\frac{1}{n}}\right]=\)
- A \(\log 2-\frac{1}{2}\)
- B \(e^{\left(\log 2-\frac{1}{2}\right)}\)
- C \(e^{\left(\frac{2 \log 2-1}{3}\right)}\)
- D \(\frac{1}{3}(2 \log 2-1)\)
Answer & Solution
Correct Answer
(C) \(e^{\left(\frac{2 \log 2-1}{3}\right)}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \quad \mathrm{I}=\lim _{x \rightarrow \infty}\left[\left(1+\frac{1}{n^3}\right)^{\frac{1}{n^3}}\left(1+\frac{8}{n^3}\right)^{\frac{4}{n^3}}\left(1+\frac{27}{n^3}\right)^{\frac{9}{n^3}} \ldots(2)^{\frac{1}{n}}\right] \\ & \Rightarrow \log \mathrm{I}=\lim _{n…
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