Question

If \[ f(x)=\frac{x^3}{1-3x+3x^2}, \] then \[ \lim_{n\to\infty} \frac{2}{n} \sum_{r=1}^{n} f\!\left(\frac{r}{n}\right) = \]

• A. \(\dfrac12\)
• B. \(\dfrac32\)
• C. \(1\)
• D. \(2\)

Hint:

For Riemann sum problems, \[ \boxed{ \lim_{n\to\infty} \frac1n \sum_{r=1}^{n} f\!\left(\frac{r}{n}\right) = \int_0^1f(x)\,dx. } \] If \[ \boxed{f(x)+f(1-x)=1,} \] then \[ \boxed{\int_0^1f(x)\,dx=\frac12.} \]

Answer 1

The Correct Option is C