<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">Spočtěte limitu:
   $$
      \lim_{x\to0_+}{\left(\log\frac1x\right)}^x.
   $$
   \newpage
   \reseni\begin{equation*}\begin{split}
      \lim_{x\to0_+}{\left(\log\frac1x\right)}^x&amp;=
      \lim_{x\to0_+}\left(\exp\left(x\log\log\frac1x\right)\right)
      \stackrel{\text{VLSF}}=\\&amp;
      \stackrel{\text{VLSF}}=
      \exp\left(\lim_{x\to0_+}\left(x\log\log\frac1x\right)\right)=\\
      &amp;=\exp\left(\lim_{x\to0_+}\left(\frac{\log\log(1/x)}{1/x}\right)\right)
      \stackrel{\text{l'H}}=\\
      &amp;\stackrel{\text{l'H}}=
      \exp\left(\lim_{x\to0_+}\left(\frac{\frac1{\log(1/x)}\cdot x\cdot
      \left(-\frac1{x^2}\right)}{-\frac1{x^2}}\right)\right)=
      \exp\left(\lim_{x\to0_+}\frac x{\log\frac1x}\right)=\\&amp;=
      \exp\left(-\lim_{x\to0_+}\left(\frac x{\log x}\right)\right) = 0.
   \end{split}\end{equation*}
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