How do you solve this limit question?

Question

limit x^(-5)lnx
as x approaches + infinity

Answer 1

= ln x / x^5
ln x -> inf
x^5 -> inf
differentiate both sides (Lagrange)

=> 1/x / 5x^4
= 1/ (5x^5)
=> 0

Or ln x increases much slower as x gets bigger, whereas x^5 increases faster, (exponentially) as it gets bigger. polynomial wins over logarithm.

The limit is 0.

Answer 2

It's too hard to write maths in here so go check out this website: http://www.dynamaths.com/cours/terminale/le_logarithme_neperien-5.html
(part 2 limites, theoreme 3)
Even though it's not in english you can take the idea, which is to use the theorem of comparison.
At the end, instead of dividing by x you divide by (x^5) but it doesn't change anything to the result.
Hope that helps a little...