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.
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 $\infty$.