As stated in the title, the problem is
Evaluate $$ \lim_{n \to \infty} \int_{-1}^{2} \frac{(1+t)^ne^{-nt}}{\frac{1}{n}+\ln{(1+nt^2)}}dt. $$
Since this is from a measure theory course, I'm supposed to solve this using the dominated convergence theorem, but I can't find an upper bound for $f_{n}$. The best I can do is a substitution $x = nt$, which yields $$ \lim_{n \to \infty} \int_{-n}^{2n} \frac{(1+\frac{x}{n})^{n}e^{-x}}{1+n\ln{(1+\frac{x^2}{n})}}, $$
and I can't find an upper bound for this whose integral from $-n$ to $2n$ (actually, on $\mathbb{R}$ but multiplied by $\chi_{[-n,2n]}$) converges.