The limit comparison test takes two different series $\sum_{i=1}^{n}a_{n}$ and $\sum_{i=1}^{n}b_{n},$ each with positive terms, then checks $\frac{a_{n}}{b_{n}} \to \ell,$ as $n \to \infty.$ Moreover, $\ell \neq 0.$
In some sources, say Thomas' Calculus, there's an additional statement about $\lim_{n \to \infty}\frac{a_{n}}{b_{n}} = \infty.$ Why is this statement usually not included in a more advanced treatment of the test, say Brannan's First Course in Mathematical Analysis?
Perhaps because it is too trivial to be mentioned. After all, if $\lim_{n\to\infty}\frac{a_n}{b_n}=\infty$, then $a_n>b_n$ if $n\gg1$ and therefore, if $\sum_{n=1}^\infty b_n$ diverges, then so does $\sum_{n=1}^\infty a_n$.