Behaviour of functions in weighted sobolev spaces

Question

If $f$ and $Df$ are in $L^2(\mathbb{R}, e^{u^2} dx)$, can we say $f(u)e^{\frac{u^2}{2}}$ is bounded. Here $Df$ distributional derivatie of $f$. That is, If $\int_{\mathbb{R}} \lvert f(u) \rvert^2 e^{u^2}du < \infty$ and $\int_{\mathbb{R}} \lvert Df(u) \rvert^2 e^{u^2}du < \infty$. 1) Can we say $f(u)e^{\frac{u^2}{2}}$ is bounded? 2) can we say $\lim_{x \rightarrow \infty} \lvert f(x) p(x) \rvert = 0$ for every real polynomial?