Evaluate the limit by first recognizing the sum as the Riemann sum for a function defined on the interval [0,1]: lim as n approaches infinity (1/n)((sqrt(1/n)+(sqrt(2/n)+(sqrt(3/n)+...............+(sqrt(n/n))

Solution:

The limit formula is the representation of the behavior of the function at a specific point and the formula analyzes that function. 

Limit describes the behavior of some quantity that depends on an independent variable, as that independent variable approaches or comes close to a particular value. 

\(\\\lim_{n\rightarrow +\varpi } (\frac{1}{n}.(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\sqrt{\frac{3}{n}}+.....+\sqrt{\frac{n}{n}})) = \frac{1}{n}\sum_{i=1}^{n}(\sqrt{\frac{i}{n}}) \\ \\By\: further\: simplification \\ \\=\int_{0}^{1}\sqrt{x}dx \\ \\Integrating\: with\: respect\: to\: x\)

\(\\=[\frac{2}{3}x^{\frac{3}{2}}]^{1}_{0} \\ \\Substituting\: the\: upper\: limit\: and\: lower\: limit \\ \\=\frac{2}{3}.1^{\frac{3}{2}} -\frac{2}{3}.0^{\frac{3}{2}} \\ \\So\: we\: get \\ \\=\frac{2}{3}\)

Therefore, the limit by first recognizing the sum as the Riemann sum for a function defined on the interval [0,1] is 2/3.

Evaluate the limit by first recognizing the sum as the Riemann sum for a function defined on the interval [0,1]: lim as n approaches infinity (1/n)((sqrt(1/n)+(sqrt(2/n)+(sqrt(3/n)+...............+(sqrt(n/n))

Summary:

\( \\=[\frac{2}{3}x^{\frac{3}{2}}]^{1}_{0} \\ \\=\frac{2}{3}.1^{\frac{3}{2}} \\ \\=\frac{2}{3}.0^{\frac{3}{2}} \\ \\=\frac{2}{3} \)