Given

N $?<>0$

we need

$>0$

so that

ln$($x$)---$Select$---<$ N $>$ N $<>$

whenever

$0<$ x $<?$ N $$ ;

that is$,$

x $=$ eln$($x$)?<>$ eN

whenever

$0<$ x $<?$ N $.$

This suggests that we take

$=$ eN$.$

If

$0<$ x $<$ eN$,$

then

ln$($x$)?<>$ ln$($eN$)=$ N$.$

By the definition of a limit$,$

lim$$x$0+$ln$($x$)=$$.$