Question on the Squeeze theorem In this theorem we consider the functions $f$, $g$ and $h$ which are defined on $\bar {\mathbb{R}}$ except possibly at $a \in \bar {\mathbb{R}}$ but could we have the limit in $a$ of these three functions equal to $ \infty$ when $f,g,h : \bar {\mathbb{R}} \to \mathbb{R}$ or do these functions must be necessary  on $\bar {\mathbb{R}} \to \bar {\mathbb{R}}$ ?
Thanks in advance.
 A: You can used one side only (it is enough). For example if for $x$ near but not at $a$ we have $f(x) \le g(x),$ and we know $\lim_{x\to a}f(x)=+\infty,$ then we can conclude that $g$ also goes to $+\infty$ as $x \to a.$
Added: Some definition of $\bar {\mathbb{R}}$ must be fixed on. There are two of these, one being the "two point" compactification in which two infinities, $-\infty,+\infty$ are adjoined to the reals, and the other in which only one unsigned infinity is adjoined. In limit contexts one could use either a two point or a one point compactification, but I think the two point is more common.
Then if the function $f: \bar {\mathbb{R}} \colon \to \mathbb{R}$ is being discussed, it is still common practice to allow $\infty$ or $-\infty$ to be the value of a limit, and in that case it just means that as $x \to a$ the values of $f(x)$ in the reals become arbitrarily large or arbitrarily small ("large negative"). However one has to read the calc book or whatever source to be certain about this usage. Some texts simply say that an infinite limit "does not exist" or is "undefined". That convention may allow some theorems to be stated without making special exceptions to them. But in my opinion, stating the limit in question is $+ \infty$ for example conveys more information than just saying the limit does not exist, and so is preferable.
