I know a question With the exact same title has an answer but it hasn’t areally answered mine so please hear me out.
I know what it means for the limit At a point to “ not exist”; the function doesn’t approach a certain value as x approaches some value. But i have seen this phrase used in 2 different contexts. One where the one-sided limits are different and
the second where function approaches infinity or negative infinity from both sides.
The notation for the second, “infinite limit, is always written out in the regular limit notation by the author, who says that it is “describing the way in which limit does not exist”. What are the differences in these 2 types of non-existences in context of the precise definition of a limit not existing?
Secondly, the author explains how The limit of a quotient cannot be computed by the quotient of the limits when the limit of the denominator is equal to zero and that of the numerator is Positive and proceeds to say that the “limit does not exist” without mentioning the limit notation used for the “infinite limits”, so I assume that he’s implying that this is the first kind of non existence. Shoudn’t this be the second type, since this is no longer an intermediate form and should clearly approach a very large positive number?
Sorry for not formatting, I post on a tablet and dont know how to format.
Best Answer
We say the the limit $\lim_{x\to a}f(x)$ is a real number $l$ when$$(\forall\varepsilon>0)(\exists\delta>0)(\forall x\in D_f):|x-a|<\delta\implies|f(x)-l|<\varepsilon,$$we say that $\lim_{x\to a}f(x)=\infty$ when$$(\forall M\in\Bbb R)(\exists\delta>0)(\forall x\in D_f):|x-a|<\delta\implies f(x)>M,$$and we say that $\lim_{x\to a}f(x)=-\infty$ when$$(\forall M\in\Bbb R)(\exists\delta>0)(\forall x\in D_f):|x-a|<\delta\implies f(x)<M.$$The limits $\lim_{x\to a^+}f(x)$ and $\lim_{x\to a^-}f(x)$ are similar, but then $|x-a|<\delta$ becomes $0<x-a<\delta$ and $-\delta<x-a<0$ respectively. In any case, $\lim_{x\to a}f(x)$ exists if and only if both limits $\lim_{x\to a^+}f(x)$ and $\lim_{x\to a^-}f(x)$ exist.
When someone says that the limit $\lim_{x\to a}f(x)$ exists, that person should let it clear whether he or she is talking about existence in $\Bbb R$ or about existence in $\Bbb R\cup\{\pm\infty\}$. Assuming that the context here is just the existence in $\Bbb R$, how can a limit fail to exist? These are the possibilities: