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

Let −∞ ≤ a < b ≤ ∞, let f and g be continuously differentiable functions defined on the interval (a, b), and let r = f/g and ρ = f /g. In [13], general “rules” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. For example, according to Proposition 1.9 in [13], one has the following: if ρ is increasing and gg > 0 on (a, b), then r ցր, which means that there is some c in [a, b] such that r is decreasing on (a, c) and increasing on (c, b). In particular, if c is either a or b, the result is that r is either increasing or decreasing on the entire interval (a, b). If one also knows whether r is increasing or decreasing in a right neighborhood of a and in a left neighborhood of b, then one can discriminate with certainty between these three patterns. Using such rules, one can generally determine ([13, 16]) the monotonicity pattern of r given that of ρ, however complicated the latter might be. Clearly, these l’Hospital-type rules for monotonicity patterns are helpful wherever the l’Hospital rules for limits are. Moreover, the monotonicity rules apply even outside such contexts, because they do not require that both f and g (or either of them) tend to 0 or ∞ at any point. In the special case when both f and g vanish at an endpoint of the interval (a, b), l’Hospital-type rules for monotonicity can be found, in different forms and with different proofs, in [1]–[4], [6]–[8], [10], and [12]–[15]. In view of what has been said here, it should not be surprising that a very wide variety of applications of these l’Hospital-type rules for monotonicity patterns were given in those papers; see also [16]. In the present note, discrete analogues of l’Hospital-type rules both for monotonicity and limits are given.