
problematic limit
hi!
i have a problem in finding limit for this funciton f (x) = x*(lnx)^23,where x > 0. i thought i will be able to use Lopital's rule, but cannot transformate funcition to uncertainity in order to use Lopital's rule. Do anyone have ideas? And I also think that there should be way to find limit without Lopital's rule, using formulas like e^x1~x.
But anyway, i have no ideas for the moment. Can anybody help?
thanks.

As x tends to zero, the given expression becomes an indeterminate of the form http://latex.codecogs.com/gif.latex?0.\infty
So convert it into an indeterminate form which allows application of L'Hospital rule .
Like this : http://latex.codecogs.com/gif.latex?...{\ln(x))}^{23}
http://latex.codecogs.com/gif.latex?...))}^{23}/(1/x)
Now consider the numerator, http://latex.codecogs.com/gif.latex?({\ln(x))}^{23} and the denominator http://latex.codecogs.com/gif.latex?(1/x)
Both terms become infinite as x tends to zero, so L'Hospital's rule is applicable.

In fact you can apply L'hopital:
(as x > 0)
by repeatedly applying L'hopital you should find:

Hello, waytogo!
Ah, Dinkydoe beat me to it . . .
Well, I'm not going to delete all this!
As given, the limit is: .
We have: . . . We can apply L'Hoptal's Rule
Apply L'Hopital: . . . We can apply L'Hoptal's Rule.
Apply L'Hopital: .
Apply L'Hopital: .
. . . . .
Apply L'Hopital: .
Apply l"Hopital: .
Apply L'Hopital: .
Therefore: .