# Calculate a hard limit

Calculate: $$\lim_{x\to+\infty} x\left( \frac{1}{x^2+1^2}+\frac{1}{x^2+2^2}+\dots+\frac{1}{x^2+x^2}\right)$$

• Transform it into a Riemann sum. Jan 15, 2014 at 19:50
• What have you tried yourself? Do you have any ideas about basic techniques for calculating limits and whether they work or not on this specific case? Jan 15, 2014 at 19:50
• How? I have tried many times to solve this question. Jan 15, 2014 at 19:51
• $x$ is a natural number? Jan 15, 2014 at 19:54
• Yes, $x$ is a natural number. Jan 15, 2014 at 19:56

$$\lim_{n \to \infty}n \Bigg( \sum_{k=1}^n {\frac {1}{n^2+k^2}}\Bigg) = \lim_{n \to \infty}\frac{1}{n}\Bigg(\sum_{k=1}^n {\frac {n^2}{n^2+k^2}}\Bigg)= \lim_{n \to \infty}\frac{1}{n}\Bigg(\sum_{k=1}^n {\frac {1}{1+(\frac{n}{k})^2}}\Bigg)$$
$$=\int_{0}^1 {\frac {dx}{1+x^2}}=\arctan(1)-\arctan(0)=\frac{\pi}{4}$$
\begin{align} & \phantom{=} \lim_{x\to+\infty} x\left( \frac{1}{x^2+1^2} + \frac{1}{x^2+2^2} + \dots + \frac{1}{x^2+x^2}\right) \\[8pt] & = \lim_{x\to\infty} \frac 1 x \left( \frac{1}{1+\frac{1^2}{x^2}} + \frac{1}{1+\frac{2^2}{x^2}} +\cdots+\frac1{1+\frac{x^2}{x^2}} \right) \\[8pt] & = \int_0^1 \frac{1}{1+w^2} \, dw = \frac \pi 4. \end{align}