lim x->0+ x^x?

Question

What is:
lim x->0+ x^x

That question has been posted on here before but the solution-

x^x = e^(xln(x)) = 1

does not explain how to get to that point. I was wondering if someone could show me the steps. This is what I have so far...
1. y = x^x
2. ln(y) = xln(x)
3. ln (y) = ln(x)/ (1/x)
4. confused

Any ideas? Thanks.

Thank you all for your posts. Kyle had the same answer as my professor gave. I am just failing to see the connection as to why he stresses that I need to solve for the limit (which is 0) before I can solve for y (which is 1). All he said is you need to solve and plug back in. Time to visit the math center! Thanks.
Go Hokies!

Answer 1

Let y = x^x

ln(y) = ln(x^x)
ln(y) = x ln(x)
ln(y) = ln(x) / (1/x)

Use L'hopital's rule:
lim as x-->0 of ln(x) / (1/x)
= lim as x-->0 of (1/x) / (-1/x^2)
= lim as x-->0 of -x
= 0

So the lim as x-->0 of ln(y) = 0
Thus the lim as x-->0 of y = e^0 = 1

Answer 2

Use L'Hospital rule:

lim x->+0; ln x = -inf

lim x->+0; 1/x = inf

lim x->0; ln(x)/(1/x) = lim x->0;(1/x)/-(1/x^2) =
= - lim x->0; (1/x)(1/x^2) =
= -lim x->0; 1/x * x^2 = -lim x->0; x = 0

Thus lim x->0 x^x = lim x->0 e^(x ln(x)) = e^0 = 1

Answer 3

ln (y) = ln(x)/ (1/x)
I guess at this point you have to use l'hospital rule

lim x->0 ln(x)/(1/x) = lim x->0 ln'(x)/(1/x)' =

lim (1/x)/(-1/x^2) = lim x->0 (-x) = 0
Therefore ln(y) approaches 0 and then y approaches 1.

Answer 4

coming near infinity: considering the two the numerator and denominator are non-end applications, the cost of the cut back of each is in simple terms the cost of the applications at x = a million that's 0. for this reason, using the quotient rule for limits might effect in a nil/0 form. coming near detrimental infinity: The spinoff of the numerator is 2x + a million whose cut back is 3 as x techniques a million. The spinoff of the denominator is 2x - a million whose cut back is likewise a million as x techniques a million. this grants a three/a million form meaning the cut back is 3. i'm assuming that as you're asking this question, which you have some understanding and which you will understand this answer.

Answer 5

x^2-1/x-1