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Here's what he says (including the preceding paragraph):

"To show in general that $f$ (where $f(x)=1/x$) approaches $1/a$ near $a$ for any $a$ we proceed in basically the same way, except that, again, we have to be a little more careful in formulating our initial stipulation. It's not good enough simply to require that $|x-a|$ should be less than $1$, or any other particular number, because if $a$ is close to $0$ this would allow values of $x$ that are negative (not to mention the embarrassing possibility that $x=0$, so that $f(x)$ isn't even defined!).

The trick in this case is to first require that $|x-a| < \frac{|a|}{2}$; in other words, we require that $x$ be less than half as far from $0$ as $a$."

Okay, so I completely get the inequality, and what he's trying to say and why, but it seems like the English "in other words" part is not an equivalent statement to the inequality.

If I were to translate the inequality to plain English, I would say, "in other words, $x$ must be less than half as far from $a$ as $a$ is from $0$." Or a more literal translation of the inequality itself, "the difference between $x$ and $a$ must be less than half that of the difference between $a$ and $0$."

Normally I would just skid right by this type of thing and assume that the textbook author just made a typo, but given that this is Spivak and it's the fourth edition and whatnot I started to wonder whether I was just misunderstanding the logic of the statement, which irks me like nothing else.

Here are the only two possible meanings of the sentence that I can discern (neither of which make sense in my view):

(1) "$x$ must be less than half as far from $0$ as $x$ is from $a$"

-or-

(2) "$x$ must be less than half as far from $0$ as $a$ is from $0$"

So suppose he means (1). So $x$ is a certain distance from $a$. Then the distance between $x$ and $0$ is less than half that distance. Even discounting the "half" specification, this statement would mean that $x$ is closer to 0 than it is to $a$, which, it seems to me, is the opposite of what that inequality is saying. Would (1) not translate to $|x|<\frac{|x-a|}{2}$?

Now suppose he means (2). So $a$ is a certain distance from $0$. Then the distance between $x$ and $0$ is less than half that distance, which also seems to me to be blatantly incorrect, because according to the inequality, there can be an $x$ that is FARTHER from zero than $a$. Would (2) not translate to $|x|<\frac{|a|}{2}$?

As I said before, I kind of doubt that I am correct about this - despite the fact that I can't think of a way that it could make sense - just because of the book that it's coming from. But I suppose there will always be typos in any book, so I'd just like to know if this is one.

(I kind of doubt many of you happen to have a copy of Spivak's lying around, but this is on page 94 and it has a nice little diagram/number line that agrees with the inequality but not the statement... it seems to me.)

Sorry for the length. Just trying to emulate Spivak's super thoroughness. :)

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    $\begingroup$ Yes, the "in other words" part is wrong. $\endgroup$ – André Nicolas Jul 13 '14 at 1:04
  • $\begingroup$ It should be: "$x$ should be less than half as far from $a$ as $a$ is from $0$." $\endgroup$ – Thomas Andrews Jul 13 '14 at 1:18
  • $\begingroup$ An alternative way to say it is that $x$ is between $a$ and $a/2$. You can't write that as $a/2<x<a$ because when $a$ negative, you don't have $a/2<a$. $\endgroup$ – Thomas Andrews Jul 13 '14 at 1:26
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Spivak corrected this phrase on p. 3 (Top) of his errata dated 2/14/2018:

Page 94, paragraph beginning "The trick in this case":

For more clarity, in the third line $x$ be less than half as far from 0 as $a$" should
be $x$ be less than half as far from $a$ as $a$ is from 0"

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