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.
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.
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.