Prove that $limxlnx$ = 0 as x approaches positive 0, given that $lim(lnx)/x$ = 0 as x approaches positive infinity?

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Answer 1 · 2017-06-23T10:43:42

We are given that:

$lim_(x rarr oo) lnx/x= 0$

Let $u=1/x$ then as $x rarr oo => u rarr 0^+$

Substituting into the above limit we get:

$lim_(u rarr 0^+) u ln(1/u)= 0$

$:. lim_(u rarr 0^+) u ln(u^(-1))= 0$

$:. lim_(u rarr 0^+) -u lnu= 0$

$:. lim_(u rarr 0^+) u lnu= 0$ QED

Prove that limxlnxlimxlnx = 0 as x approaches positive 0, given that lim(lnx)/xlim(lnx)/x = 0 as x approaches positive infinity?