Consider a sequence of functions $f_n : (0,\infty) \rightarrow \Bbb R$ defined by
$$f_n(x)=\frac{n}{n + x + nx^2}$$
Show that $f_n(x)\le f_{n+1}(x)$ for all $n \in \Bbb N$ and $x \in (0,\infty)$. Hence, compute
$$\lim_{n\to \infty}\int_{(0,\infty)}\frac{n}{n+x+nx^2}$$
$$f_n(x)=\frac{1}{1 + x^2 + \frac{x}{n}}\lt \frac{1}{1 + x^2 + \frac{x}{n+1}}=f_{n+1}(x)$$ and $$f_n(x) \lt \frac{1}{1+x^2} \text{ for all $n\ge 1, x \gt 0$ }$$ Since $f_n(x)\to f(x) = 1/(1+x^2)$ pointwise and $f$ is integrable, monotone convergence gives $$\lim_{n\to \infty}\int_0^\infty\frac{n}{n+x+nx^2} = \int_0^\infty\frac{1}{1+x^2} = \frac{\pi}{2}$$