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We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators K by calculating them in the finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators 
$\langle x_{1}|K|x_{0}\rangle $.
We study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem
where xu is the (unique) maximum point of |u|, 
$$\left\{ {\matrix{ {-(\Delta _p + \Delta _{q(p)})u = \lambda _p \vert u(x_u) \vert ^{p-2}u(x_u)\delta _{x_u}} & {{\rm in}} & \Omega \cr {u = 0} \hfill \hfill \hfill & {{\rm on}} & {\partial \Omega ,} \cr } } \right.$$
$\delta _{x_{u}}$ is the Dirac delta distribution supported at xu,
and λp > 0 is such that
$$\mathop {\lim }\limits_{p\to \infty } \displaystyle{{q(p)} \over p} = Q\in \left\{ {\matrix{ {(0,1)} & {{\rm if}} & {N < q(p) < p} \cr {(1,\infty )} & {{\rm if}} & {N < p < q(p)} \cr } } \right.$$
$$\min \left\{ {\displaystyle{{{\rm \Vert }\nabla u{\rm \Vert }_\infty } \over {{\rm \Vert }u{\rm \Vert }_\infty }}:0 \ne u\in W^{1,\infty }(\Omega )\cap C_0(\bar{\Omega })} \right\} \les \mathop {\lim }\limits_{p\to \infty } (\lambda _p)^{1/p} < \infty .$$
