As stated in the title, for a given $p \in [1,+\infty)$ and $f \in L^{p}(0,+\infty)$, I want to find the value of $$ \lim_{n \to \infty} \int_{0}^{\infty} \frac{\tan{\frac{\pi}{2(1+n^{p}x)}}}{n\ln{(1+x)}}f(x)dx. $$
This is an exercise from a functional analysis exam, and I'm fairly certain that this can be solved using the uniform boundeness principle and its corollary (i.e. define $A_{n}:L^{p}(0,+\infty) \to \mathbb{C}$, $A_{n}f = \int_{0}^{\infty} \frac{\tan{\frac{\pi}{2(1+n^{p}x)}}}{n\ln{(1+x)}}$, show that $A_{n}$ are uniformly bounded, i.e. $\sup_{n \geq 1} ||A_{n}|| < +\infty$, and $\lim_{n \to +\infty} A_{n}f$ for all $f \in S$, $S$ such that $\overline{\mathcal{Lin}S} = L^{p}(0, \infty)$, guess what the "limit" operator should be based on its values on $S$, and apply the principle).
However, $A_{n}$ are not bounded on $L^{p}(0,+\infty)$, not even well-defined, because for $f(x) = e^{-x} \in L^{p}(0, \infty)$ for all $p \in [1, +\infty)$, the integral diverges for all $n \in \mathbb{N}$, because the $e^{-x}$ behaves like $1$ and $0$, and the fraction behaves like $C/x^2$ at $0$, which diverges. However, the integral would definitely be defined for a function that goes to $0$ very rapidly at both $0$ and $+\infty$. Is there a good subspace of $L^{p}(0, +\infty)$ for which one could actually find this limit using the aforementioned method?