91Ó°ÊÓ

In calculus, **limits** serve as the foundational building block for more complex concepts such as derivatives and integrals. Limits help us understand the behavior of functions as they approach a certain point. Here, the limit describes what happens to the Riemann sum as the number of subintervals increases infinitely.
When evaluating \( \lim_{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{n}{n^{2}+k^{2}} \), the goal is to grasp the behavior of our sum as \( n \), the number of subintervals, approaches infinity. This process helps us transition from finite sums to the continuous area under a curve.
The critical step is recognizing that as \( n \) becomes very large, the sum begins to resemble a definite integral, providing a powerful method for evaluating such sums efficiently using calculus techniques.

The **definite integral** is a central concept in calculus that gives the accumulated total of a function over a specific interval. In our problem, the definite integral is formed as we transition from a Riemann sum.

• The given integral \( \int_0^1 \frac{1}{1+x^2} \, dx \) emerges as the limit of the Riemann sum, providing the exact area under the curve of the function \( f(x) = \frac{1}{1+x^2} \).


• The definite integral can be thought of as the sum of infinitely many tiny quantities, offering a precise method to calculate areas and solve various real-world problems.

A crucial part of solving these problems lies in the ability to connect Riemann sums, which approximate the integral, to the definite integral itself, which represents the exact value in the limit as \( n \to \infty \).
The evaluated integral results in \( \tan^{-1}(x) \) from \( 0 \) to \( 1 \), demonstrating how easily we can move from sums to precise calculations of areas.

**Subintervals** play a pivotal role in approximating areas under curves through Riemann sums. By dividing an interval \( [0, 1] \) into \( n \) equal parts, each segment's width is determined by \( \Delta x = \frac{1}{n} \).

• This division allows us to approximate the area under a curve by summing up the areas of rectangles with widths \( \Delta x \) and heights determined by the function's value at specific points.


• In Riemann sums, the choice of sample points, often the right endpoints \, \( k \Delta x \), significantly influences the approximation.

As \( n \) increases, these subintervals shrink, improving our sum’s approximation accuracy and leading to its equivalence to a definite integral in the limit.
This process demonstrates how we bridge discrete sums with continuous functions through a limit approach.