Product of two functions with respectively an infinite limit and a limit equal to 0 at some point.

This situation is frequently encountered with logarithms and exponentials; see subsection 5.4. We give here more simple examples.

  1. Take $ f(x)=x$ and $ g(x)=1/x$ ; then we have

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x)=+\infty$   and$\displaystyle \qquad \underset{x \rightarrow +\infty}{\lim} g(x)=0,$    

    hence the computation of the limit at $ +\infty$ of $ f(x)g(x)$ shows an undeterminate case. We have:

    $\displaystyle f(x)g(x)=x \cdot \frac 1x = 1,$    

    whence

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x)g(x)=1.$    

  2. Take $ f(x)=x^4$ and $ g(x)=1/x^2$ ; then we have

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x)=+\infty$   and$\displaystyle \qquad \underset{x \rightarrow +\infty}{\lim} g(x)= 0.$    

    Here we have

    $\displaystyle f(x)g(x)=\frac{x^4}{x^2}=x^2$    

    whence

    $\displaystyle \underset{x \rightarrow +\infty}{\lim} f(x)g(x)= +\infty.$    

Noah Dana-Picard 2007-12-28