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I was attending a lecture in computational fluid dynamics when an equation popped up with a variable that could go to infinity. My mind wandered and I started thinking of the following, completely unrelated case:

$$ x = \lim_{a \to 0} a \cdot \infty$$

What would x be in this case? I believe that the limit above would be treated as zero by definition, meaning the answer would be zero. Intuition says something different though, as a "goes to" zero but never really is zero. It's a bit of a silly question, but I'm curious how you guys look at this.

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How do you define "$\infty$"? –  Jp McCarthy Sep 11 '13 at 13:38
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Usually we don't actually have $\infty$ sitting there in our problem. Instead we have problems of the form $\lim_{x \to 0}(f(x) g(x))$, where $f(x)$ goes to zero and $g(x)$ goes to infinity. Calculus courses deal with this (and related problems) under the heading "indeterminate forms". –  GEdgar Sep 11 '13 at 14:00

1 Answer 1

up vote 2 down vote accepted

We can make sense of this limit in the extended real number line. In the extended real number line, we leave $0 \times \infty$ undefined. However, $a \times \infty$ is defined for $a \neq 0$. If $a > 0$, it is $+ \infty$, and if $a < 0$, it is $-\infty$.

Thus, in the above limit, the right-hand limit goes to $+ \infty$, and the left-hand limit to $-\infty$. Thus the limit does not exist in the extended real number line. However, if we define this limit in the projective real line, where $+\infty = -\infty$, then the limit exists, and is equal to $\infty$.

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