One-forms and infinitesmally small distances

Question

I have recently been studying a bit of differential geometry and have become somewhat familiar with integrating volume forms. In physics (and also maths) often times, we say things like $dx$ is an infinitesimally small distance, and that as $\Delta x \rightarrow 0$ we end up with $dx$. I was wondering what the link between $dx$ being a one-form and $dx$ being an infinitesimally small distance is. Thank you!