limit of ax-1x as x approaches 0


For a>0, we have


Recall that ax=exlna. Thus,

limx0ax-1x =limx0exlna-1x

Let t=xlna. Then t0 as x0. Therefore,

limx0ax-1x =(lna)limt0et-1t

The formula from the corollary is useful for proving that ddxax=axlna. On the other hand, once this fact is known, the corollary is easily proven via l’Hôpital’s rule (

limx0ax-1x =limx0axlna1
Title limit of ax-1x as x approaches 0
Canonical name LimitOfdisplaystylefracax1xAsXApproaches0
Date of creation 2013-03-22 17:40:21
Last modified on 2013-03-22 17:40:21
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 5
Author Wkbj79 (1863)
Entry type Corollary
Classification msc 32A05