Let f , be a continuous function in [ 0 , 1 ] , then lim n → ∞ ∑ j = 0 n n 1 f ( n j ) is

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Let $f$, be a continuous function in $[0,1]$, then $\lim_{n \rightarrow \infty} \sum_{j=0}^n \frac{1}{n} f\left(\frac{j}{n}\right)$ is

MathematicsDefinite IntegrationWBJEEWBJEE 2020

Options:

A $\frac{1}{2} \int_{0}^{\frac{1}{2}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$
B $\int_{\frac{1}{2}}^{1} \mathrm{f}(\mathrm{x}) \mathrm{dx}$
C $\int_{0}^{1} f(x) d x$
D $\int_{0}^{\frac{1}{2}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$

Solution:

1716 Upvotes Verified Answer

The correct answer is: $\int_{0}^{1} f(x) d x$

\(\lim _{n \rightarrow \infty} \sum_{i=0}^{n} \frac{1}{n} f\left(\frac{i}{n}\right)\)
Let \(1 / n \rightarrow d x\)

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