Question

\(\lim _{n \rightarrow \infty} \frac{3}{n}\left\{1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}+\ldots+\sqrt{\frac{n}{n+3(n-1)}}\right\}\)
(A) does not exit
(B) is 1
(C) is 2
(D) is 3

Answer 1

Ans : (C)
Hint \(: \operatorname{Lt}_{n \rightarrow \infty} \frac{3}{n} \sum_{r=0}^{n-1} \sqrt{\frac{1}{1+3\left(\frac{r}{n}\right)}}\)
$$
=3 \int_{0}^{1} \frac{d x}{\sqrt{1+3 x}}=2[\sqrt{1+3 x}]_{0}^{1}=2
$$

Answer 2

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Answer 3

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