how to solve the limit of an integral with an unknown function? With $f \in L^1(\mathbb{R})$, solve :
$$
\lim_{n\rightarrow \infty} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} f(x)\tan^n(x)\,\mathrm{d}x
$$
But I have no idea where to start. I think I have to do an integration by parts and apply a convergence theorem, but I can't understand how it's possible to calculate $f(x)$ when it's not defined.

Notice that for all $x\in\left(-\frac{\pi}{4},\frac{\pi}{4}\right)$ we have that $\tan x\in(-1,1)$, and so in particular also that $\tan^n x\in(-1,1)$. This means that for all $x\in\left(-\frac{\pi}{4},\frac{\pi}{4}\right)$,
$$\left\lvert f(x)\tan^n x\right\rvert\leq\left\lvert f(x)\right\rvert.$$
Since $f$ is $L^1$, we can apply the Dominated Convergence Theorem, which yields that
$$\lim_{n\to\infty}\int_{\left(-\frac{\pi}{4},\frac{\pi}{4}\right)}f(x)\tan^n x~\mathrm{d}\lambda(x)=\int_{\left(-\frac{\pi}{4},\frac{\pi}{4}\right)}f(x)\cdot0~\mathrm{d}\lambda(x)=0.$$
