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All the definitions I can find of a limit (with functions from R to R) define something like:

"as x approaches a, f(x) approaches L"

Where x is treated as a variable that is quantified over in the definition.

Whereas many of these books then go on to use expressions of the form:

"as g(x) approaches a, f(x) approaches L"

without generalizing the definition appropriately.

Two questions:

  1. what on earth makes this seem unproblematic to the authors? I'm guessing that the way I view things makes this use of notation seem more problematic than it is.

  2. What is the appropriate formal defintion of the limit of f(x) as g(x) approaches a, where f:S->R and S is a subset of R.

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    If $g$ is invertible in a neighborhood of $a$ then there are no problems here since $f$ is locally defined as a function of $g$. To be honest, though, I don't know any examples where it isn't already clear that $f$ can be written as a function of $g$; can you clarify?2011-03-15
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    Are you saying "lim f(x) as g(x)->a" is short for "lim f(g(x)) as x -> g^-1(a)"? If so, why does g need to be invertible in a neighborhod, rather than just at the point? (I mean, the second limit may not exist, but shouldn't that just tell us, then, that the first limit does not exist either?)2011-03-15

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