What is the difference between the expressions
$$ \lim\sup\left|\frac{a_{n+1}}{a_n}\right| \ \ \ \ \ \mbox{and} \ \ \ \ \ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| $$
specifically with respect to the statement of the Ratio Test outlined in Ross' text?
Suppose : $ b_n = \left|\frac{a_{n+1}}{a_n}\right| $ where $n = 1, 2, \dots $, $(a_n \neq 0)$.
Then $$\lim\sup_{n\rightarrow \infty} b_n = \inf_{ n \rightarrow \infty} \left(\sup_{k \geq n} b_k\right).$$
While the other one is $\lim_{n \rightarrow \infty} b_n$.
Note that (as mentioned in the comments) : while taking inf and supremum, the values $\infty$ and $-\infty$ are allowed to be taken, while in the case of limits it is not.
When limit exists it will coincide with $\lim\sup$, the latter always exists but the former may not.
Check also definition of $\lim\inf$.