Spočtěte limitu:
   $$
      \lim_{x\to0_+}{\left(\log\frac1x\right)}^x.
   $$
   \newpage
   \reseni\begin{equation*}\begin{split}
      \lim_{x\to0_+}{\left(\log\frac1x\right)}^x&=
      \lim_{x\to0_+}\left(\exp\left(x\log\log\frac1x\right)\right)
      \stackrel{\text{VLSF}}=\\&
      \stackrel{\text{VLSF}}=
      \exp\left(\lim_{x\to0_+}\left(x\log\log\frac1x\right)\right)=\\
      &=\exp\left(\lim_{x\to0_+}\left(\frac{\log\log(1/x)}{1/x}\right)\right)
      \stackrel{\text{l'H}}=\\
      &\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)=\\&=
      \exp\left(-\lim_{x\to0_+}\left(\frac x{\log x}\right)\right) = 0.
   \end{split}\end{equation*}