find the limit

Question

lim  (ln(x^-1))^x

x->0+

Answer 1

Let y = (ln(1/x))^x; lny = xln(ln(1/x)); lim lny as x→0⁺ = lim as x→0⁺ ln(ln(1/x))/(1/x) = (∞/∞ Lopital's) =

lim as x→0⁺ 1/ln(1/x)·x·(1/x)'/(1/x)' = lim as x→0⁺x/ln(1/x) = 0

So, lim as x → 0⁺ (ln(1/x))^x = e⁰ = 1_