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For example:

the limit of $e^t$ as $t$ approaches infinity is simply infinity.

but the limit of $e^{t^2}$ cannot be found because it explodes? Is this due to the fact that there are differing types of infinity and the second limit produces a non-"countable" infinity?

see referenced question

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    Tha$n$ks, Hans! MathOverflow doesn't see$m$ to have this problem...2010-10-20

3 Answers 3

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The limit can be found. What are we saying when we assert that $lim_{t\to\infty} f(t) = \infty$, with $f:\mathbb{R} \to \mathbb{R}$? What does it mean?

It means, by definition, that given M >0 there exists \varepsilon >0 such that if t>\varepsilon then f(t) > M. Simply this. In (not so rigorous) words, for arbitrary large values of the parameter $t$ the function assumes arbitrary large values.

It is easy to see that the limit you want is indeed $\infty$ (in the sense just described). This can be seen noticing that $e^t < e^{t^2}$ for t>1.

I think that this question is not related to the linked one. For me, they are different things.

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To take a different perspective on it (if this is what you're looking for), there are limits that can't be found; the limit of $1/x$ as $x\rightarrow 0$ is one example. The limit doesn't exist because the limit coming from the right and from the left are different: one is $\infty$ and one is $-\infty$. I guess you could say that $1/x$ "explodes" at 0.

It's important to distinguish the type of infinity you're talking about, too. The "countable and uncountable" infinities you mention are cardinalities: they measure the number of elements in sets. The infinities you get with limits of functions like these are just limits of real numbers, and don't describe sets at all. You could call them "uncountable" since $\mathbb{R}$ is uncountable, I suppose, but then there wouldn't be a "countable" infinity in between $\infty$ and the real numbers. Basically, "infinity" can mean a lot of different things.

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Given your mentioning of the word "countable" (and looking at the link), I think that the "differing types of infinity" that you are thinking of are different infinite cardinalities (i.e. different possible infinite sizes of sets). This notion of infinity is not the same as the notion of infinity that is used when one says that a limit approaches infinity. (Of course, they are related at a basic level, in that both allude to the idea of being larger than any finite number, but at a more technical level, they are different notions.)

Ronaldo in his answer has given an explanation of what it means for $\displaystyle \lim_{t\to\infty} f(t)$ to equal $\infty$, and this is just as valid for $e^{t^2}$ as it is for $e^t$.

As Dan Ramras notes in one of his comments, the rate at which $e^{t^2}$ approaches infinity is much greater than the rate at which $e^t$ approaches infinty (which is turn much greater than the rate at which $t$ approaches infinity, for example). In terms of Ronaldo's answer, this means that for a given $M$, the $\epsilon$ that you have to choose to be sure that e^{t^2} > M when t > \epsilon is smaller than in the corresponding situation for $e^t$. But both quantities approach infinity as $t$ does.

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    @SamB: Thanks for this.2010-10-20