mathop {lim }limits_{n to infty } left{ {frac | vedclass

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(A) The given limit is $\mathop {\lim }\limits_{n 	o \infty } \left( {\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}} + \frac{3}{{{n^2}}} + \dots + \frac{n}{{{

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$1/2$

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(A) The given limit is $\mathop {\lim }\limits_{n 	o \infty } \left( {\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}} + \frac{3}{{{n^2}}} + \dots + \frac{n}{{{n^2}}}} ight)$.We can factor out $\frac{1}{{{n^2}}}$ from the sum:$= \mathop {\lim }\limits_{n 	o \infty } \frac{1}{{{n^2}}} (1 + 2 + 3 + \dots + n)$.Using the formula for the sum of the first $n$ natural numbers,$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$:$= \mathop {\lim }\limits_{n 	o \infty } \frac{n(n+1)}{2{n^2}}$.Simplify the expression:$= \mathop {\lim }\limits_{n 	o \infty } \frac{n^2 + n}{2n^2} = \mathop {\lim }\limits_{n 	o \infty } \left( \frac{1}{2} + \frac{1}{2n} ight)$.As $n 	o \infty$,$\frac{1}{n} 	o 0$:$= \frac{1}{2} + 0 = \frac{1}{2}$.

$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}} + \frac{3}{{{n^2}}} + \dots + \frac{n}{{{n^2}}}} \right\}$ is

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