Pomocí Taylorovych rozvojù vypoètìte:
   $$
      \lim_{x\to0}\frac{\cos x-e^{-x^2/2}}{x^4}.
   $$
   \reseni
   Taylorovy polynomy stupnì 4 pro obì funkce z èitatele jsou:
   \begin{align*}
      \cos x      &= 1 - \frac{x^2}2 + \frac{x^4}{24} + o(x^4)\\
      e^{-x^2/2}  &= 1 - \frac{x^2}2 + \frac{x^4}8 + o(x^2)
   \end{align*}
   Poèítáme limitu:
   \begin{multline*}
      \lim_{x\to0}\frac{\cos x-e^{-x^2/2}}{x^4}=\\
      =\lim_{x\to0}\biggl(\frac{1-(x^2/2)+(x^4/24)+o(x^4)}{x^4}-\\
      - \frac{1-(x^2/2)+(x^4/8)-(x^6/48)+(x^8/384)+o(x^4)}{x^4}\biggr)=\\
      =\lim_{x\to0}\frac{-x^4/12 + x^6/48 - x^8/384 + o(x^4)}{x^4}=-\frac1{12}\
   \end{multline*}
