3
$\begingroup$

Suppose $g:\mathbb{R} \rightarrow [0,\infty)$ is a strictly increasing function such that $\lim g(x) = \infty$ as $x \rightarrow \infty$. Suppose $h:\mathbb{R} \rightarrow [0,\infty)$ is a strictly decreasing function such that $\lim h(x) = 0$ as $x \rightarrow \infty$. Consider the product function $f(x)=g(x)h(x)$. Is it possible to construct such $g,h$ such that $f$ does not have a limit as $x \rightarrow \infty$?

Even better, can such an $f$ be constructed if we insist that $g(x)=x$?

  • 0
    Generally the way to do these kinds of questions is to construct f first. What have you tried?2010-12-13
  • 0
    If you do not want to have your product diverging either, the only way is $g$ having infinitely many saddle-points, says my intuition. Is this possible and sufficient?2010-12-13
  • 0
    Do you consider a limit that is equal to $\infty$ to exist or not to exist? (In my opinion, the latter).2010-12-13

3 Answers 3