Consider a complex number $z = x + i y$ and functions $f,g : \mathbb{H} \to \mathbb{C}$ (where $\mathbb{H}$ is the upper half-plane i.e. complex numbers whose imaginary part is greater equal to zero). Actually we can ask that $f,g$ are modular forms for $SL_2(\mathbb{Z})$. Then one can define the Petersson inner product as $$ \langle f, g \rangle := \int_{\mathbb{H}/SL_2(\mathbb{Z})} f(z) \overline{g(z)} y^{2k}\frac{dx dy}{y^2} $$ as is defined in equation (2.3) in this paper. Now, we read in section 3 of the same paper that if $f,g$ are weakly holomorphic modular forms (that is meromorphic modular forms all of whose poles are contained at cusps) then the inner Petersson inner product will generally diverge. I understand that this is the case since we end up integrating (as far the interesting imaginary part of $z$ is concerned) all the way up to infinity.
I would like to ask for a simple, down to earth explicit example where this divergence is apparent and what a simple way to regularize such a divergence would be. I just need to get some intuition really.