L'Hospital-type rules for monotonicity and limits: Discrete case

TL;DR

This paper extends l'Hospital-type rules to discrete functions, establishing conditions under which ratios of functions defined on integers exhibit monotonicity and limit behaviors, similar to continuous cases.

Contribution

It introduces discrete analogs of l'Hospital-type rules for monotonicity and limits, applicable to functions on integer intervals, filling a gap in discrete analysis.

Findings

Discrete l'Hospital rules for monotonicity established

Conditions for ratio behavior on integer domains derived

Analogous limit rules for discrete functions provided

Abstract

Assuming that a "derivative" ratio rho:=f'/g' of the ratio r:=f/g of differentiable functions f and g is monotonic (that is, rho is increasing or decreasing), it was shown in previous papers that then r can switch at most once, from decrease to increase or vice versa. In the present paper, "discrete" versions of such l'Hospital-type rules for monotonicity (as well as "discrete" versions of l'Hospital's rules for limits) are obtained, for functions f and g defined on an interval of integers.