Homework Help

What is limit ? lim x^x  x->0 x>0

user profile pic

d1nk | eNotes Newbie

Posted June 9, 2013 at 2:59 PM via web

dislike 1 like

What is limit ?

lim x^x 

x->0

x>0

Tagged with limit hopital, math

1 Answer | Add Yours

Top Answer

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 9, 2013 at 3:21 PM (Answer #1)

dislike 2 like

You need to use l'Hospital's theorem such that:

`lim_(x->x_0) f^g = lim_(x->x_0) e^(ln(f^g))`

Using the relation above, yields:

`lim_(x->0,x>0) x^x = lim_(x->0,x>0) e^(ln(x^x))`

Using logarithmic power identity yields:

`lim_(x->0,x>0) e^(ln(x^x)) = lim_(x->0,x>0) e^(x ln x)`

`lim_(x->0,x>0) e^(x ln x) = e^(lim_(x->0,x>0) x ln x)`

You need to evaluate the limit of exponent such that:

`lim_(x->0,x>0) x ln x = lim_(x->0,x>0) (ln x)/(1/x) = oo/oo`

The indetermination oo/oo requests for you to use l'Hospital's theorem again yields:

`lim_(x->0,x>0) (ln x)/(1/x) = lim_(x->0,x>0) ((ln x)')/((1/x)')`

`lim_(x->0,x>0) ((ln x)')/((1/x)') = lim_(x->0,x>0) (1/x)/(-1/x^2) = lim_(x->0,x>0) -x = 0`

Hence, evaluating the limit yields:

`lim_(x->0,x>0) e^(x ln x) = e^0 = 1`

Hence, evaluating the limit using l'Hospital's theorem, yields `lim_(x->0,x>0) x^x = 1` .

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes