The value of lim _{n rightarrow infty} frac{1 | vedclass

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(C) The given expression is $\lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n (k^2 x)$.We can rewrite this as $x \lim _{n \rightarrow \infty} \

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$\frac{x}{3}$

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(C) The given expression is $\lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n (k^2 x)$.We can rewrite this as $x \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n (\frac{k}{n})^2$.Using the definition of the definite integral as the limit of a Riemann sum,$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n f(\frac{k}{n}) = \int_0^1 f(t) dt$.Here,$f(t) = t^2$,so the expression becomes $x \int_0^1 t^2 dt$.Evaluating the integral: $x [\frac{t^3}{3}]_0^1 = x (\frac{1}{3} - 0) = \frac{x}{3}$.

The value of $\lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n (k^2 x)$ is

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