In this paper we deal with a connection between the upper Kuratowski limit of a sequence of graphs of multifunctions and the upper Kuratowski limit of a sequence of their values. Namely, we will study under which conditions for a graph cluster point $(x,y)\in X\times Y$ of a sequence $\{GrF_n:n\in\omega\}$ of graphs of lower quasi-continuous multifunctions, $y$ is a vertical cluster point of the sequence $\{F_n(x):n\in\omega\}$ of values of given multifunctions. The existence of a selection being quasi-continuous on a dense open set (a dense $G_\sigma$-set) for the topological (pointwise) upper Kuratowski limit is established.