# Limit of probability density function as random variable approaches +/- infinity

Consider a complex-valued function $\Psi(x,t)$ such that $|\Psi|^2$ is a probability density function for $x$ (for any time $t$).

In his introductory Quantum Mechanics book, David J Griffiths writes that the limit of the expression $$\Psi^* \frac{\partial \Psi}{\partial t}-\frac{\partial \Psi^*}{\partial t} \Psi^*$$ as $x \rightarrow \infty$ and as $x \rightarrow -\infty$ must be $0,$ stating that $\Psi$ would not be normalizable otherwise. However, in a footnote, he mentions that "A good mathematician can supply you with pathological counterexamples, but they do not come up in physics; for the wave function always goes to $0$ at infinity."

What are some counter-examples to this claim?

More generally, if $f$ is a function, and $$\int_{-\infty}^{\infty}f(x)dx=1$$ then when may we assume that $\lim_{x\to\pm\infty}f(x) = 0$? Or, for positive integers $n$, that $\lim_{x\to\pm\infty} f^{(n)}(x)=0$?

$$f(x)=\begin{cases} 1-2x & x \in [0,1/2] \\ 2x+1 & x \in [-1/2,0] \\ 0 & \text{otherwise}\end{cases}.$$
Geometrically, this is a triangle with height $1$ and width $1$ centered at zero.
$$g(x)=\sum_{n=1}^\infty f(n^2(x-n))$$
This is a sequence of separate triangles with height $1$ and width $1/n^2$. The total area is then $\sum_{n=1}^\infty \frac{1}{2n^2}$, which is finite, but $g$ does not have a limit at $+\infty$. (Of course we can make it fail at $-\infty$ as well.)