TS EAMCET · Maths · Limits
\(\lim _{n \rightarrow \infty} \frac{(2 n(2 n-1) \ldots(n+2)(n+1))^{1 / n}}{n}=\)
- A \(\int_0^1 \log x d x\)
- B \(\int_0^1 x \log x d x\)
- C \(\int_0^1(x+1) \log (x+1) d x\)
- D \(\int_0^1 \log (1+x) d x\)
Answer & Solution
Correct Answer
(D) \(\int_0^1 \log (1+x) d x\)
Step-by-step Solution
Detailed explanation
\(\lim _{n \rightarrow \infty} \left( \prod_{k=1}^{n} \frac{n+k}{n} \right)^{1/n}\) \( = \lim _{n \rightarrow \infty} \left( \prod_{k=1}^{n} \left(1+\frac{k}{n}\right) \right)^{1/n}\) Let \(L\) be the limit. Then…
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