The value of lim _{n rightarrow infty} left[ | vedclass

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(B) The given limit is $L = \lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{n}{n^{2}+r^{2}}$.Dividing the numerator and denominator by $n^{2}$, we g

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$\frac{\pi}{4}$

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(B) The given limit is $L = \lim _{n ightarrow \infty} \sum_{r=1}^{n} \frac{n}{n^{2}+r^{2}}$.Dividing the numerator and denominator by $n^{2}$, we get:$L = \lim _{n ightarrow \infty} \sum_{r=1}^{n} \frac{1/n}{1+(r/n)^{2}}$.Using the definition of a definite integral as the limit of a sum, $\lim _{n ightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r}{n}ight) = \int_{0}^{1} f(x) dx$, where $f(x) = \frac{1}{1+x^{2}}$.Therefore, $L = \int_{0}^{1} \frac{1}{1+x^{2}} dx$.Evaluating the integral, we get $L = [	an ^{-1} x]_{0}^{1}$.$L = 	an ^{-1}(1) - 	an ^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$.

The value of $\lim _{n \rightarrow \infty} \left[ \frac{n}{n^{2}+1^{2}} + \frac{n}{n^{2}+2^{2}} + \ldots + \frac{n}{n^{2}+n^{2}} \right]$ is

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