1. Understand the limit:
We need to evaluate: \[ \lim_{n \to \infty} \left( \frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \cdots + \frac{1}{5n} \right) \]
2. Recognize the sum as a Riemann sum:
The given expression can be written as: \[ \lim_{n \to \infty} \sum_{k=1}^{4n} \frac{n}{n^2 + k^2} \] This resembles the Riemann sum for an integral.
3. Convert to an integral:
Divide numerator and denominator by \( n^2 \): \[ \frac{n}{n^2 + k^2} = \frac{1/n}{1 + (k/n)^2} \] The sum becomes: \[ \lim_{n \to \infty} \sum_{k=1}^{4n} \frac{1}{n} \cdot \frac{1}{1 + (k/n)^2} \] This is the Riemann sum for: \[ \int_{0}^{4} \frac{1}{1 + x^2} dx \]
4. Evaluate the integral:
\[ \int_{0}^{4} \frac{1}{1 + x^2} dx = \tan^{-1} x \Big|_{0}^{4} = \tan^{-1} 4 - \tan^{-1} 0 = \tan^{-1} 4 \] However, the upper limit should be 2 (since \( k \) goes up to \( 2n \) when the last term is \( \frac{1}{5n} \)): \[ \int_{0}^{2} \frac{1}{1 + x^2} dx = \tan^{-1} 2 \]
Correct Answer: (C) \( \tan^{-1} 2 \)