`lim_(x->0^+) (x^x - 1)/(ln(x) + x - 1)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule...

`lim_(x->0^+) (x^x - 1)/(ln(x) + x - 1)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

The denominator has limit `-oo+0-1=-oo.`

The numerator has limit 0 (I'll prove this below).

Therefore there is no indeterminacy, `0/oo=0.`

(zero is the answer).

Now prove that `x^x->1` when `x->0^+:`

`x^x = (e^lnx)^x = e^(x*lnx).`

`x*lnx = lnx/(1/x),` limit of this exists by l'Hospital's Rule: `oo/oo,` everything is differentiable for x>0, `f'/g' = (1/x)/(-1/x^2) = -x -> 0.`