Compute $\lim_{n \to \infty} \int_0^1 \frac{n dx}{(1+nx)^2(1+x+x^2)}$

Question

Compute $$\lim_{n \to \infty} \int_0^1 \frac{n dx}{(1+nx)^2(1+x+x^2)}$$

I initially tried to apply theorems that allow me to move the limit under the integral, until I realized that the integral approaches 1 while the function approaches 0 as $n \to \infty$. Open to any suggestions at this point. Thanks.

Answer 1

Hint: do the change of variable $y = nx$, so that your integral becomes $$ \int_0^n \frac{1}{(1+y)^2} \cdot \frac{1}{1+y/n + y^2/n^2}\, dy = \int_0^\infty f_n(y)\, dy, $$ with $$ f_n(y) := \frac{1}{(1+y)^2} \cdot \frac{1}{1+y/n + y^2/n^2}\, \chi_{[0,n]}(y). $$ Then use the Dominated Convergence Theorem.