# Limits of a function on a space larger than the domain of the function

I've recently been reviewing some calculus, and came across a question. A proof of the power rule for natural numbers (in Stewart's Calculus book) uses the following telescoping trick, given $a \in \mathbb{R}$ and $x$ a variable ranging over $\mathbb{R}$:

$(x^{n} - a^{n}) = (x-a)(x^{n-1} + x^{n-2}a + \cdots + xa^{n-2} + a^{n-1})$.

Then we compute the derivative, and at one point we get the equality

$\lim_{x \rightarrow a}\frac{x^{n} - a^{n}}{x - a} = lim_{x \rightarrow a}(x^{n-1} + x^{n-2}a + \cdots + xa^{n-2} + a^{n-1})$.

The only way I could make sense of this was to think of $\frac{x^{n} - a^{n}}{x - a}$ and $(x^{n-1} + x^{n-2}a + \cdots + xa^{n-2} + a^{n-1})$ as technically different functions; the first is a restriction of the second to the domain $\mathbb{R} - \{a\}$. Then the above statement is then o.k. because the $\epsilon - \delta$ definition of limits of functions in metric spaces makes an allowance for it, i.e. in:http://www.proofwiki.org/wiki/Definition:Limit_of_Function_(Metric_Space).

My question: is the proofwiki definition standard? If it is then I guess limits of functions can be defined on larger spaces than the functions themselves are defined on. A brief internet search hasn't provided too much, and the books i have don't talk about exactly this point. Thanks for any advice, and feel free to move this to meta or some other place that might be more fitting.

Yes, it is standard when taking the limit $\displaystyle\lim_{x\to a}f(x)$ to only consider $x$ in a deleted neighbourhood of $a$. As you point out, this is necessary when considering the definition of a derivative (i.e. the differential quotient isn't defined at $x = a$).