Consider a point p in a manifold with coordinates [tex]x^\alpha[/tex] and another point nearby with coordinates [tex]x^\alpha + dx^\alpha[/tex] where dx are infinitesimal or arbitrarily small. Suppose we have a function f on this manifold. Then we can write [tex]df=f(x^\alpha + dx^\alpha)-f(x^\alpha)=\partial_\mu f dx^\mu[/tex] Notationally this is identical to the expression for the d operator acting on the scalar function f to give the 1-form [tex]df=\partial_\mu f dx^\mu[/tex] where the dx^mu here are the coordinate basis of the 1-forms at p. Surely the similarity in notation is meaningful in some way. But how? Now I can easily repeat back to you the definition of a tangent vector as a map from functions on the manifold to real numbers, and the definition of 1-forms as the dual space to the tangent space. But I can't see the conceptual relationship to small displacements. Furthermore, since tangent vectors at a point p can also be thought of as directional derivatives to curves passing through p, visually it is tangent vectors which seem to me more like small displacements, rather than their dual the 1-forms.