This is from my textbook.
the textbook says it makes no sense when $b=\infty$, I think what it want to show us is $\Delta x$ could also be $\infty$ because of $b$, but as $n$ is also $\infty$, it still makes a little sense that $\Delta x$ could still be very small ($\frac{\infty}{\infty}$)?
Your post does not really contain a question to answer. But like @vrugtehagel said, you should consider $\infty$ represent a concept rather than a number. For example, say $\infty := \ 'redness'$
Then saying something like $redness - 2$ simply does not mean anything. Or, $ \frac{redness}{redness} %$, for that matter.
Luckily, as you probably already learned from your calculus book, our tools allow us to compute limits involving infinity. For example, if we define
$$ \int_a^\infty {f(x)dx} = \lim_{c\to \infty} \int_a^c{f(x)dx}$$
We can easily evaluate the indefinite integral.