Is this 1 statement true about limits [Calculus Question About Limits] Is this 1 statement true about limits?
If the left side and the right side of the function is approaching infinity or negative infinity then the limit is unbounded. If the limit is unbounded it also means there is no limit.
I tried asking people but I got different results. And everyone was arguing, some said that the statement is true and others said that the statement if false. So which one is it as different people have different answers to my question. thanks
 A: (This answer assumes a calculus context, as the Question is tagged "calculus".  In a real analysis context, some of what is written below is modified (because one works over the extended reals).)
In calculus, infinity and minus infinity are ideas related to a process.  Positive infinity means a process that produces values that, for any bound you pick, the process eventually exceeds and forever remains above that bound.  (It may bounce up and down across that bound for a while, but eventually it crosses one last time and is forever greater than the bound.)  This has to be true for any bound.  Then we say that the process goes to infinity.  For instance $1/x$ as $x \rightarrow 0^+$ describes a process (the outputs of $1/x$ as positive $x$s shrink towards $0$) that produces ever greater values.  For any bound we pick, we can find a small enough $x$ such that $1/x$ is bigger than the bound and any smaller positive $x$ is above that bound.
The case with negative infinity is analogous.  Here, the process produces outputs that eventually are forever below any bound.  Is it eventually forever less than $-1$?  Yup.  Eventually forever below $-10$?  Yup. Eventually forever below $-10^{10^{10^{10}}}$?  Yup.  Eventually forever below (some stupendously huge negative number)?  Yup. ...
So that's what positive and negative infinity are in the context of limits : ideas capturing a process that eventually forever overcomes any bound (overcoming upward for positive infinity and overcoming downward for negative infinity).
Then by definition, if a function approaches positive or negative infinity in a limit, its values are unbounded.  But one does not say "the limit is unbounded".  If both sides eventually increase without bound, the limit is infinity.  If both sides decrease without bound, the limit is minus infinity.  A limit is either a finite number, infinity, or minus infinity because a limit is a completed process.  A limit cannot be "unbounded" because there is no process left to be increasing or decreasing.  Once you have the limit, the process is over.
When the value of a limit is a finite number, then the limit exists.  In all other cases, the limit does not exist.  If you have an infinite limit, you pretend that $\infty$ or $-\infty$ are numbers and write either "$\lim_{x \rightarrow c} f(x) = \infty$" or "$\lim_{x \rightarrow c} f(x) = -\infty$" (or any one-sided variant), but the limit does not exist.  However, the particular behaviour of going to positive or negative infinity is both common and useful, so we have special language and notation to describe these special kinds of nonexistent limit.
So, if the process is unbounded, the limit does not exist.  It may be one of the special infinite types that we write as if we were pretending that $\infty$ and $-\infty$ were numbers instead of ideas.  But even in those cases, those are special kinds of nonexistent limit.
