In axiomatics of the alternative set theory (AST), construction of some hyperreal structures is considered rather often. These structures are based on horizons that are initial segments or, in different terms, segments of the class of natural numbers. As a rule, they are wider than the class of all finite natural numbers. Contrary to the classical situation, monotone sequences of elements of such hyperreal structure may have no limits even if they are bounded. Necessary and sufficient conditions for a base segment of the structure when such paradoxical situation is impossible have been obtained already earlier. In this paper, we obtain conditions on the growth rate or decrease rate of monotone sequence with a limit within the given hyperreal structure. The relation between the limit and precise upper and lower boundaries of classes is investigated. Causes and mechanisms of limitlessness are examined. It is shown that harmonic series in a hyperreal structure converge if and only if the base segment of the structure is definable set-theoretically or, in other words, has an exact upper boundary.