In this paper a multidimensional Tauberian theorem is proved that establishes a connection between the behavior of a generalized function in a cone and the behavior of its Laplace transform in the neighborhood of zero in the tube domain over the cone. Here it is assumed that the Laplace transform has nonnegative imaginary part or, more generally, bounded argument. The theorem is used to illuminate sufficient conditions for the existence of an angular limit of holomorphic functions of bounded argument. An example is constructed of a holomorphic function with bounded nonnegative imaginary part in $T^{R_+^2}$, having a limit over a countable set of rays coming into the origin, but without an angular limit. In addition, a number of theorems on the existence of quasi-asymptotic limits of the solutions of multidimensional convolution equations are proved, and examples are considered of finding quasi-asymptotic limits of fundamental solutions of hyperbolic operators with constant coefficients, as well as of passive systems. The quasi-asymptotic limit of a fundamental solution of the system of equations governing a rotating compressible fluid is found, and similarly for other systems. Bibliography: 10 titles.