Extending "as x approaches a" to "as g(x) approaches a"

Question

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.

Answer 1

The delta-epsilon definition of the limit generalizes in a straightforward way. Suppose that for each $\epsilon > 0$, there exists $\delta(\epsilon) > 0$ such that $|g(x)-a|<\delta(\epsilon) \implies |f(x)-L|<\epsilon$. Then we say that $f(x)$ approaches $L$ as $g(x)$ approaches $a$. Though it might be more appropriate to say "whenever $g(x)$ approaches $a$", since this could happen for more than one value of $x$. For instance, $\sin(x)$ approaches $0$ whenever $\cos(x)$ approaches $1$.

Answer 2

I would interpret the statement that f(x) approaches L as g(x) approaches a to mean that for every $\epsilon > 0$ there exists $\delta > 0$ such that for all x, $0 < |g(x) - a| < \delta$ implies $|f(x) - L| < \epsilon$.

Note that x need not be a real variable in order to apply this definition, though doubtless examples of that type come to mind. For example, sin(x) tends to 0 as cos(x) approaches 1, but the converse is not true.