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$lim_{n \to \infty} \frac{1}{n} [1 + \frac{n^{2}}{n^{2} + 1^{2}} + \frac{n^{2}}{n^{2} + 2^{2}} + \dots + \frac{n^{2}}{n^{2} + (n - 1)^{2}}]$
is equal to

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detailed solution

Correct option is C

limn→∞ 1n1+n2n2+12+n2n2+22+⋯+n2n2+(n−1)2=limn→∞ 1n∑r=0n−1 n2n2+r2=limn→∞ 1n∑r=0n−1 11+rn2=∫01 dx1+x2=tan−1 x01=π4

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limn→∞ 1n1+n2n2+12+n2n2+22+⋯+n2n2+(n−1)2is equal to

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$lim_{n \to \infty} \frac{1}{n} [1 + \frac{n^{2}}{n^{2} + 1^{2}} + \frac{n^{2}}{n^{2} + 2^{2}} + \dots + \frac{n^{2}}{n^{2} + (n - 1)^{2}}]$
is equal to