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.


1 Answer 1


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 $x = a$).


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