In this paper, we study the asymptotic behavior of the v-number of a Noetherian graded filtration I={I[k]}k≥0 of a Noetherian N-graded domain R. Recently, it was shown that v(I[k]) is periodically linear in k for k≫0. We show that all these linear functions have the same slope, i.e. [Formula presented] exists, which is equal to [Formula presented], where α(I) denotes the minimum degree of a non-zero element in I. In particular, for any Noetherian symbolic filtration I={I(k)}k≥0 of R, it follows that [Formula presented], the Waldschmidt constant of I. Next, for a non-equigenerated square-free monomial ideal I, we prove that v(I(k))≤reg(R/I(k)) for k≫0. Also, for an ideal I having the symbolic strong persistence property, we give a linear upper bound on v(I(k)). As an application, we derive some criteria for a square-free monomial ideal I to satisfy v(I(k))≤reg(R/I(k)) for all k≥1, and provide several examples in support. In addition, for any simple graph G, we establish that v(J(G)(k))≤reg(R/J(G)(k)) for all k≥1, and v(J(G)(k))=reg(R/J(G)(k))=α(J(G)(k))−1 for all k≥1 if and only if G is a Cohen-Macaulay very-well covered graph, where J(G) is the cover ideal of G.