Limit of function scaling when parameter tends to zero

Question

Given any $\lambda>0$, consider the dilation operator $D_{\lambda}f(x)=f(\lambda x)$. Does it make any sense to consider the limit $\lambda\rightarrow 0$? Given that the function spaces I am interested in are the classical Lebesgue spaces $L^p(\mathbb{R}^n)$ and the Schwartz class $\mathcal{S}(\mathbb{R}^n)$, is there any hope to give a meaning to this limit? For a suitably well-behaved function $f$, the intuition suggests that this yields the constant function $f(0)$ (this also suggests convergence to Dirac's delta, at least in distributional sense). I cannot see how to deal with Lebesgue $p$-integrable (equivalence classes of) functions, much less with tempered distributions $(\mathcal{S}'(\mathbb{R}^n))$. Any suggestions?