The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre polynomial $L_n$, and the weights $w_1,...,w_n$ are chosen according to $w_i=\frac{1}{x_i (L_n'(x_i))^2}$.

The intuition is that if $f$ is polynomial of degree at most $2n-1$ then the approximation is exact; In general, the approximation error (as in any Gauss quadrature) is known to be given by $$E_n(f) = \frac{(n!)^2}{(2n)!} f^{(2n)}(\xi)$$ for some $\xi \in (0,\infty)$. My question is simply the following:

Is there function $f$ that is smooth in $(0,\infty)$, such that the the approximation error $E_n(f)$ does not go to zero as $n\to \infty$?

My feeling is that if $f$ derivative grows very fast, or, say, $f$ has infinite derivative at the end point 0, e.g., $f(x)=x \log x$, then maybe $\xi$ is very close to zero and the error will not improve as $n$ grows, but I am no expert in this field. Any comment will be welcomed.