calculus - Function whose integral over $[0, infty)$ is finite, but the limit of the function is $>0$

Let $t \in \mathbb{R},f:\mathbb{R} \rightarrow \mathbb{R}:t \mapsto f(t)$.

$f$ is required to have:

1. $\displaystyle\lim_{t \rightarrow \infty} f(t) = L$, where $L>0$ (could be $\infty$, so the limit exists, only it could be $\infty$)




2. $\displaystyle\int_{0}^{\infty} f(t) dt < \infty$

Is that possible to construct such $f$?

Note:

$f$ can be any function continuous or discontinuous.

I found the discussion here which is involving "tent" function but in that example, the limit does not exist.

Note 2: I have edited the title. Thank you for your fedback !!!

Thank you for any answers or comments.