limh→0(a^x-1)/h为何能化简成In(a)

Question

limh→0(a^x-1)/h 为何能化简成In(a)

Answer 1

设a^h-1=y,a^h=1+y,h=log_a(1+y)=ln(1+y)/lna
h→0时y→0
lim(h→0)(a^h-1)/h
=lim(y→0)y/[ln(1+y)/lna]
=lna*lim(y→0)y/ln(1+y)
=lna*lim(y→0)1/[1/y*ln(1+y)]
=lna*lim(y→0)1/{ln[(1+y)^(1/y)]}
已知lim(y→0)(1+y)^(1/y)=e,故
原式=lna*1/lne=lna