\begin{frame}
  \frametitle{Precise Definition of Infinite Limits - Example}
  
  \preciseposinf
  \pause\smallskip

  \begin{exampleblock}{}
    Proof that $\lim_{x\to 0} \frac{1}{x^2} = \infty$.
    \pause\bigskip
    
    Let $M$ be a positive number. \pause
    We look for $\delta$ such that
    \begin{talign}
      \text{if } \quad 0 < |0-x| < \delta \quad\text{ then }\quad \frac{1}{x^2} > M
    \end{talign}
    \pause
    We have:
    \begin{talign}
      \frac{1}{x^2} > M  
      \mpause[1]{\;\;\iff\;\; 1 > M\cdot x^2} 
      \mpause[2]{\;\;\iff\;\; \frac{1}{M} > x^2}
      \mpause[3]{\;\;\iff\;\; \sqrt{\frac{1}{M}} > |x|}
    \end{talign}
    \pause\pause\pause\pause
    Thus $\delta = \sqrt{1/M}$.\pause\quad
    $\text{If } \quad 0 < |0-x| < \sqrt{1/M} \quad\text{ then }\quad \frac{1}{x^2} > M$.
  \end{exampleblock}  
\end{frame}