EXAMPLE $6$ Evaluate

lim$$x $0+$x ln$($x$3).$

SOLUTION The given limit is indeterminate because, as

x $0+,$

the first factor $($x$)$ approaches while the second factor

ln$($x$3)$

approaches

$$$.$

Writing

x $=1/(1/$x$),$

we have

$1/$x $$

as

x $0+,$

so l'Hospital's Rule gives

lim$$x $0+$x ln$($x$3)=$lim$$x $0+$

ln$($x$3)1/$x

$=$lim$$x $0+$

$3/$x$1/$x$2$

$=$lim$$x $0+$

$$$$

$=$

$0$

$.$