Convergent Improper integral whose integrand tends to a non zero finite limit as x tends to infinity.

Define a function $ f $ as the following graph shows :

It is possible to define $ f $ explicitly, but that's not a big deal.

Its graph is formed of triangles centered at integers, each one centered at $ n\in\mathbb{N} $, has a base $ B_{n}=\frac{1}{n^{2}} $, and an altitude $ h=2 $, which means has an area $ \mathcal{A}_{n}=\frac{B_{n}\times h}{2}=\frac{1}{n^{2}} \cdot $

Then $$ \int_{0}^{+\infty}{f\left(x\right)\mathrm{d}x}=\sum_{n=1}^{+\infty}{\mathcal{A}_{n}}=\sum_{n=1}^{+\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2}}{6} $$

$ \int_{0}^{+\infty}{f\left(x\right)\mathrm{d}x} $ converges, $ f\geq 0 $, but $ \lim\limits_{x\to +\infty}{f\left(x\right)}\neq 0 $ since $ \left(\forall n\in\mathbb{N}^{*}\right),\ f\left(n\right)=2 \cdot $