Prove that lim(x→0) x(ln x)^n = 0 for positive integers n

[trap]

Proving an indeterminate form

Prove for all positive integers n that $$\lim_{x\rightarrow 0}x({lnx})^n=0$$

Thanks for any help.

 

[shmoe]

That's not an integral.

Do you know l'Hopital's rule? Combine it with induction.

 

[trap]

oh...i didn't think of induction
i kept doing it by l'hopital's rule and it came out infinity/infinity all the time.
Thanks for the advice

 

[dextercioby]

It's futile to use L'Hôpital's rule (you can't get a reasonable expression for

$$\frac{d^{k}(\ln x)^{n}}{dx^{k}}$$

)

Do a substitution:

$$x=e^{-v}$$

The result is immediate.It's like comparing exp & a finite polynomial.Since "n" is fixed,the factor $(-1)^{n}$ bears no relevance...

Daniel.

 

[Hurkyl]

L'Hôpital's rule + induction works fine for me... just like Jameson said.

 

[shmoe]

It's like comparing exp & a finite polynomial.

and how do you know what happens in this comparison if you aren't familiar with it? enter l'Hôpital...