Infinite limit at infinity.

  1. $ x_0 = +\infty \; , \; l=+\infty.$

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x) = +\infty \Longleftrightarrow \forall A>0, \exists B>0 \; \vert \; x>B \Longrightarrow f(x)>A.$    

    See Figure fig Infinite limit of a function at infinity.
    Figure 5: Infinite limit of a function at infinity.
    \begin{figure}\centerline
\mbox{\epsfig{file=InfLimInfty.eps,height=4cm}}\end{figure}
  2. $ x_0 = +\infty \; , \; l=-\infty.$

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x) = -\infty \Longleftrightarrow \forall A<0, \exists B>0 \; \vert \; x>B \Longrightarrow f(x)<A.$    

  3. $ x_0 = -\infty \; , \; l=+\infty.$

    $\displaystyle \underset{x \rightarrow -\infty}{\lim} f(x) = +\infty \Longleftrightarrow \forall A>0, \exists B<0 \; \vert \; x<B \Longrightarrow f(x)>A.$    

  4. $ x_0 = -\infty \; , \; l=-\infty.$

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x) = -\infty \Longleftrightarrow \forall A<0, \exists B<0 \; \vert \; x<B \Longrightarrow f(x)<A.$    



Noah Dana-Picard 2007-12-28