mathop {Limit}limits_{n to infty } frac{1}{n} | vedclass

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(C) The given expression is $S = \mathop {Limit}\limits_{n 	o \infty } \frac{1}{n} \sum_{r=0}^{3n-3} \sqrt{\frac{n}{n+r}}$.This can be rewritten as $

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(C) The given expression is $S = \mathop {Limit}\limits_{n 	o \infty } \frac{1}{n} \sum_{r=0}^{3n-3} \sqrt{\frac{n}{n+r}}$.This can be rewritten as $S = \mathop {Limit}\limits_{n 	o \infty } \frac{1}{n} \sum_{r=0}^{3n-3} \frac{1}{\sqrt{1 + \frac{r}{n}}}$.Using the definition of the definite integral as the limit of a sum,$\mathop {Limit}\limits_{n 	o \infty } \frac{1}{n} \sum_{r=0}^{kn} f(\frac{r}{n}) = \int_0^k f(x) dx$.Here,$f(x) = \frac{1}{\sqrt{1+x}}$ and the upper limit is $k = \frac{3n-3}{n} 	o 3$ as $n 	o \infty$.Thus,$S = \int_0^3 \frac{1}{\sqrt{1+x}} dx$.Evaluating the integral: $S = [2\sqrt{1+x}]_0^3$.$S = 2\sqrt{1+3} - 2\sqrt{1+0} = 2\sqrt{4} - 2(1) = 2(2) - 2 = 4 - 2 = 2$.

$\mathop {Limit}\limits_{n \to \infty } \frac{1}{n} \left[ 1 + \sqrt {\frac{n}{n + 1}} + \sqrt {\frac{n}{n + 2}} + \sqrt {\frac{n}{n + 3}} + \dots + \sqrt {\frac{n}{n + 3(n - 1)}} \right]$ has the value equal to

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