Let $f:\mathbb{R}\rightarrow\mathbb{C}$ be an arbitrary smooth function that vanishes outside a compact set.
For $\lambda>0$, we define the parametrised integral :
$$I(\lambda):=\int_{-\infty}^{+\infty}\sqrt{\lambda}e^{-i\lambda\frac{x^2}{2}}f(x)dx$$
Does $I(\lambda)$ admit a finite limit when $\lambda\rightarrow+\infty$ ? So I either need to prove that this limit exists for every $f$ with the same hypotheses, or I can find a counterexample of a function $f$ such that $I(\lambda)$ does not converge when $\lambda\rightarrow+\infty$.