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[My personal website],
[OSU (work) website].
$\lim_{x\to 0} |A(x)| \leq \lim_{x\to 0} |B(x)| = 0$.
But we cannot take the limit of A before we know it exists! Really, what was being proved was $|A(x)| \leq |B(x)|$, and only then can we take the limit because $B(x)$ to zero. This would be enough to lose a point or two in a beginning real analysis class, and ought to at least earn a raised eybrow in a more advanced class.Some of these were found by my students, so thanks, though I didn't keep track of who told me about what. I also didn't manage to mark down some issues from the beginning of the book (I marked it down but can't find it), so don't think there are no comments on the first part, I just lost them or forgot them.