How do asymptotes relate to boundedness?
If a function has a vertical asymptote, then it will be unbounded above or below or both in any interval that contains the asymptote.
If a function has an oblique asymptote, then it will be unbounded above or below in at least one of
A continuous function that is unbounded above or below or both in a finite interval has a vertical asymptote.
A continuous function need not has asymptotes in order to be unbounded in
A discontinuous function does not need to have asymptotes in order to be unbounded in a finite interval. Consider the function
This function is unbounded both above and below in any non-trivial interval.
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