Is it possible that two sequences ${a_n \over b_n} \to 1$ but $| a_n - b_n | \to \infty$? This question occured to me while reading the Wikipedia page on the prime counting function, $\pi(n)$. It's known that ${\pi(x) \over x/\log x} \to 1$, but the page provided a table of values for their differences up to $10^{20}$ and it appeared to be growing without bound. This seemed counterintuitive; as the ratio approaches one, shouldn't the difference approach zero?
And the thought just occured to me that ${x \over x+1} \to 1$ while $x - (x+1) = -1$ but this still isn't a diverging difference.
Yes. Take $a_n =n$, and $b_n = n + \sqrt{n}$, for instance.