Question

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

Answer 1

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