Why does the limit of (n^n)*(x^(n^2)) approach 0 as n approaches infinity?

[fk378]

Homework Statement

I want to find the limit as n approaches infinity of (n^n)*(x^(n^2)), and

Homework Equations

My teacher told us to look at the log. 0<x<1. Also, since the log of this goes to -infinity, then the original limit in question goes to 0.

(1) why is looking at log valid, if log is only defined for positive numbers
(2) how did we figure out that the limit goes to 0 by looking at the log function?

 

[Dick]

(1) if 0<x<1 what's not positive here? (2) if you can show that the log of that quantity goes to -infinity, then you will have shown the original quantity goes to zero. Numbers whose logs are huge negative numbers are positive numbers close to zero. Listen to your teacher.