lim x->0 x^(sinx)

May 2010
129
0
WI - USA
Good afternoon all. I am getting back into calculs after a 5 year hiatus, and am finding that I seem to have forgotten some of the finer points.

I was wondering if someone could walk me through the follwoing:

lim, x->0, x^(sin(x))
Thanks much
 
Dec 2009
1,506
434
Russia
+

lim x^sinx = lim e^(sinx)lnx = e^lim((sinx)lnx =e^0=1

lim (x-->0+) sinx*lnx = 0, why is that?

Try l'Hopital on ln(x)/(1/sin(x)). What does that give you? Remember lim sin(x)/x=1.
 

Ted

Feb 2010
240
64
China
It goes to 0^0 which is indeterminate.
Put y=x^(sin x)
--> ln(y) = (sin x) ln(x)
Find the limit of ln(y) and let it be a.
then the answer for your question is e^a.