Evaluate the limit.
![\[ \lim_{x \to 0^+} \left( x^{x^x} - 1 \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f6251743d8501e9047842cfd441b02cc_l3.png "Rendered by QuickLaTeX.com")
We write 
and use the continuity of the exponential (and use the notation 
),
![\begin{align*} \lim_{x \to 0^+} \left( x^{x^x} - 1 \right) &= -1 + \lim_{x \to 0^+} e^{x^x \log x} \\[9pt] &= -1 + \exp \left( \lim_{x \to 0^+} x^x \log x \right) \\[9pt] &= -1 + 0 \\[9pt] &= -1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-277adc1b2885e3f0cea36d041985a1cf_l3.png "Rendered by QuickLaTeX.com")
We know 
(and hence 
) since 
by Example #3 on page 302 of Apostol.
Evaluate the limit.
We write and use the continuity of the exponential (and use the notation ),
We know (and hence ) since by Example #3 on page 302 of Apostol.