Q:

Why we can't majorate it $\int_{1}^{\infty} e^{-\frac{1}{y^{2}}(x-\frac{1}{y})^{2}}dx$

Consider $\displaystyle \int_{1}^{\infty} e^{-\frac{1}{y^{2}}(x-\frac{1}{y})^{2}}dx$, where $0 < y < 1$. It's easy to show that for $y = 1$ integral majorating and it's equal to $\int_{0}^{\infty} \exp(-x^{2})dx$. But the task said that we can't do that. 
Why we can't majorate it by integral doesn't depend of $y$?

A:

You can't do that because
$$
\sup_{y\in(0,1)}\exp\left[-{1\over y^2}\left(x-{1\over y}\right)^2\right]=1,
$$
attained for $y=1/x$.

