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?