limn →∞ (1)/(n)[1+(n^2)/(n^2+1^2)+(n^2)/(n^2+2^2)+…+(n^2)/(n^2+(n-1)^2)] is equal to (A) (π)/(2)

step 1 in this problem we have to find limit of n tends to infinity 1 over n 1 plus n square upon n squ

Limn →∞...

$\lim _{n \rightarrow \infty} \frac{1}{n}\left[1+\frac{n^{2}}{n^{2}+1^{2}}+\frac{n^{2}}{n^{2}+2^{2}}+\ldots+\frac{n^{2}}{n^{2}+(n-1)^{2}}\right]$
is equal to
(A) $\frac{\pi}{2}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{4}$
(D) $\frac{\pi}{6}$

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Key Concepts

Riemann Sum Approximation

This concept involves approximating the area under a curve by dividing the interval into equal subintervals and summing the product of the function value at specific points and the subinterval width. In the context of limits, a sum can be seen as a Riemann sum that approaches a definite integral as the number of subdivisions tends to infinity.

Definite Integrals

A definite integral is the limit of a Riemann sum as the number of partitions increases indefinitely. It represents the exact accumulation of quantities described by a continuous function over a specified interval. Recognizing when a limit of a sum can be expressed as a definite integral is crucial in transitioning from discrete summation to continuous analysis.

Variable Substitution in Limits

This concept refers to rewriting elements of a sum using a scaled variable (often normalizing the index by a factor related to the number of terms) to reveal the structure of a Riemann sum. Such substitution simplifies the limit process and helps identify the corresponding function being integrated over a fixed interval.

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