If f is a continuous positive-valued function defined on the closed interval from a to x and if k[subscript 0] is greater than 0, then lim[subscript k[right arrow]0[superscript +] [integral][superscript x] [subscript a] f (t)[superscript k-k[subscript 0]] dt= [integral][superscript x] [subscript a] f (t)[superscript -k[subscript 0] dt. This generalizes a result of Finlayson describing ln(x) as a certain limit. Applications include new descriptions of arcsin(x) and arctan(x) as limits. This note could find use as an enrichment material in an introductory calculus course in the unit(s) introducing the Riemann integral and its applications.