Home » Blog » Compute the limit of the given function

Compute the limit of the given function

Evaluate the limit.

    \[ \lim_{x \to 0^+} \left( x^{x^x} - 1 \right). \]


We write x^{x^x} = e^{x^x \log x} and use the continuity of the exponential (and use the notation e^x = \exp x),

    \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*}

We know \lim_{x \to 0^+} x^x \log x = -\infty (and hence \exp \left( \lim_{x \to 0^+} x^x \log x \right) = 0) since \lim_{x \to 0^+} x^x = 1 by Example #3 on page 302 of Apostol.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):