I need some pointers trying to solve the following:

> 
> Calculate
>  $$\lim\_{n\to \infty}\int\_\mathbb{R} e^{-nx^2+x}dx$$
> 
> 
> 

This is what I have: (I assume $n\in \mathbb{N}\setminus \{ 0\})$

1. Define $f\_n: \mathbb{R} \to [0,+\infty[ =e^{-nx^2+x}$
2. Each $f\_n$ is Lebesgue-measureable (how do you prove this?)
3. Since $f\_n$ is not increasing I'll resort to dominating convergence. I start looking for a certain $g:\mathbb{R}\to \mathbb{R}$ such that $|f\_n(x)|\leq g(x)$ and $\int\_\mathbb{R} g <+\infty$.

I noted that $-nx^2+x$ reaches its maximum for $n=1$ such that $e^{-nx^2+x} \leq e^{-x^2+x}$ However is this integrable? And how should i do this? Should I use a different bound?