$\lim_{n \rightarrow \infty}x_n= (-1)^nn$ My notes state:
$x_n= (-1)^nn$
satisfies: $x_n \to \infty $ as $n \to \infty$
however to me $x_n$ appears to alternate between positive and negative values of infinitismally large magnitude, 
so why does my textbook describe its limit as $x_n \to \infty$ ?
 A: The sequence is $-1,2,-3,4,-5,\ldots$. If one is working in the extended reals, one may talk about $-\infty$ and $+\infty$. In this case, the sequence goes to neither. However, $|x_n|\to+\infty$.
A: [Your textbook might say that $x_n \to \infty$ and $x_n \to \infty$, but you are right. The sequence is divergent and the $x_n$'s have alternating sign.] In general then I would not write as your book has done.
However, you might look at the books definition of what $... \to \infty$ means. I guess that the book might include in this notation the case where the limit is as in this sequence.

Edit: After looking at the authors webpage and looking at chapter 6 page 9, the authors definition of $f(x) \to \infty$ as $x\to a$ is that $\lvert f(x)\lvert$ can be made as large as possible. Hence for all this to be consistent, there doesn't appear to be a mistake in chapter 19 on sequences.
Note that sometimes we might like to be able to distinguish between $f(x) \to \infty$ and $f(x) \to -\infty$. 
A: It really depends on definitions. 
There are two ways to "compactify" the real line. One is to add two points, $+\infty$ and $-\infty$. The other way is to add one point at infinity, $\infty$. The latter is, in a sense, more canonical - every space has a "one-point" compactification, while only and "ordered space" has a two-point compactification.
Alternatively, you might say that $x_n\to\infty$ if $|x_n|\to +\infty$.
I really dislike the usage of $\infty$ to mean $+\infty$ for this reason, so I agree with the book's usage. It might be confusing that $+\infty$ is different from $\infty$, but infinity is confusing.
