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JEE Main Mathematics question on Limits Continuity and Differentiability

lim⁡n→∞1n∑r=12nrn2+r2\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{2n} \frac{r}{\sqrt{n^2 + r^2}} equals
  1. A.
    1+51 + \sqrt{5}
  2. B.
    5−1\sqrt{5} - 1
  3. C.
    −1+2-1 + \sqrt{2}
  4. D.
    1+21 + \sqrt{2}
Show answer and solution

Correct option: B

Solution

Convert to a Riemann sum by writing r/n = x and 1/n as dx. The sum becomes the integral from 0 to 2 of x/√(1+x²) dx (since r goes from 1 to 2n, r/n goes from 0 to 2).

Evaluating: ∫₀² x/√(1+x²) dx = [√(1+x²)]₀² = √5 - √1 = √5 - 1.

Previous-year question, JEE Main.

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