In this website (http://www.jirka.org/spivak-errata.html), it mentions that the book erroneously claims that if $$\lim_{x\to a} |A(x)| \le \lim_{x\to a} |B(x)| = 0$$ then $$\lim_{x\to a} |A(x)| = 0.$$ without showing that the limit exists. But doesn't the limit exist by Squeeze theorem? It's bounded by 0 on one side and bounded by $B(x)$ on the other.
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In order to apply the squeeze theorem in the first place, you need
$$0\le|A(x)| \leq |B(x)|.$$
Then you can consider the limit as $x\rightarrow a$ of both $0$ and $|B(x)|$ and use the squeeze theorem to conclude
$$\lim_{x\rightarrow a} |A(x)| = 0.$$
I don't know the context of this book, but it seems like the author of this website is saying that the author of this book is not first making this assumption clear.
