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I'm having trouble with considering this limit:

$\lim_{c\rightarrow0}\int_{1}^{\infty}\frac{c}{x}dx$

It is almost like writing $\lim_{c\rightarrow0}(c\infty)$, but maybe not quite the same.

Does the limit $\lim_{c\rightarrow0}\int_{1}^{\infty}\frac{c}{x}dx$ exist? Is it 0? It would appear to be zero...

But if we use the epsilon-delta definition of limit then we fail...

Should I be using some "rule" like L'Hopital's rule? If so, I don't know which one to use...

Can we bring c outside the integrand? If so, why? If not, why?

$\lim_{c\rightarrow0}(\lim_{k\rightarrow\infty}\int_{1}^{k}\frac{c}{x}dx)$

Should I be considering this one? If so, I don't know how to progress... I assume you can't just swap the limits because we have to be "careful" here as opposed to "usual".

Maybe even it doesn't make sense to ask for the first limit. Help please?

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$\int_1^\infty\frac{c}{x}dx$ doesn't exist for any $c\ne0$ so the question is nonsensical. –  anon Apr 15 '12 at 10:34
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I just spoke to someone. There are notational issues here... some people say the integral "diverges". Some people say the integral "does not exist". Others say "It converges to infinity". They all mean the same thing. My limit doesn't exist because it is not a limit of real numbers (or you could think about it in terms of sequences... same stuff happening). Apparently if we consider the limit on the projected real line then the limit makes sense and would be at the point infinity. –  Adam Rubinson Apr 15 '12 at 11:01
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If you write that out in more detail, it will be an acceptable answer (you can answer yourself). In particular, note that $\int_1^y \frac{c}{x}dx \to \infty$ if $y \to \infty$, so if you calculate the integral first, the limit would, indeed, be $\infty$. On the other hand, $\int_1^\infty \lim_{c \to 0} \frac{c}{x} dx = 0$, which is why this is interesting. –  Johannes Kloos Apr 15 '12 at 12:02
    
I agree with most of what you say. Except, strictly speaking, the "limit of integrals" you mention would not be infinity. No such limit exists, namely because it does not make sense to talk about such limit (in R). This is because you would have to consider the sequence (infinity, infinity,...), which is NOT a sequence in R tending to infinity. But in the extended real line it does make sense to talk about such a sequence, so the limit does exist in that space, and is infinity in that space. –  Adam Rubinson Apr 15 '12 at 13:21
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