  We introduce a method to produce bounds for the non secant defectivity of an
arbitrary irreducible projective variety, once we know how its osculating
spaces behave in families and when the linear projections from them are
generically finite. Then we analyze the relative dimension of osculating
projections of Grassmannians, and as an application of our techniques we prove
that asymptotically the Grassmannian $\mathbb{G}(r,n)$, parametrizing
$r$-planes in $\mathbb{P}^n$, is not $h$-defective for $h\leq
(\frac{n+1}{r+1})^{\lfloor\log_2(r)\rfloor}$. This bound improves the previous
one $h\leq \frac{n-r}{3}+1$, due to H. Abo, G. Ottaviani and C. Peterson, for
any $r\geq 4$.
