Prove that

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

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