What kind of a formula is this and does it belong to combinatorics?

Question

$$ x^n-y^n $$

Hi I have a limit to calculate with this formula and I don't know where it belongs,I only know the binomial combination.Is this formula related to combinatorics?

$$\lim_{x \to 1} \frac {x^n-1}{x^m-1} $$

$$\lim_{x \to 1} \frac{x^n-1-n(x-1)}{(x-1)^2} $$

I have 2 limits and I do not understand where $x^n-y^n$ come from. I've searched my textbooks and I couldn't find anything about this.I know that this is special factoring but I haven't found a proof of this type. Can someone help me and explain to me how these 2 powers are related to special factoring ?

$$ x^n-y^n = (x-1)*(x^{n-1}+x^{n-2}+...+1) $$

$$ y^n=1 $$

How did this formula appeared? I don't understand this solution and this solution is related to the 2 limits!

Answer 1

$$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\dots+y^{n-1})$$ (you can check it by expanding the right hand side: almost all terms cancel pairwise), hence $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+x^{n-3}+\dots+1),$$ so that $$\lim_{x\to1}\frac{x^n-1}{x^m-1}=\lim_{x\to 1}\frac{x^{n-1}+x^{n-2}+\dots+1}{x^{m-1}+x^{m-2}+\dots+1}=\frac nm$$ (the special case $m=1$ will be used at the end of the calculation below), and $$\lim_{x\to1}\frac{x^n-1-n(x-1)}{(x-1)^2}=\lim_{x\to1}\frac{x^{n-1}+x^{n-2}+x^{n-3}+\dots+1-n}{x-1}$$ $$=\lim_{x\to1}\frac{x^{n-1}-1}{x-1}+\lim_{x\to1}\frac{x^{n-2}-1}{x-1}+\dots+\lim_{x\to1}\frac{1-1}{x-1}$$ $$=(n-1)+(n-2)+\dots+0=\frac{(n-1)n}2.$$

Answer 2

Difference of Two Squares

The formula $x^n-y^n$ is a difference of two squares. For example, if $n=2$ we can write $$x^2-y^2 = (x-y)(x+y)$$ The generalized difference of two squares for any two $n$th power is $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$$ writing this as a sum, we get $$x^n-y^n = (x-y)\sum_{k=0}^{n-1}x^{n-1-k}y^k$$

Note that when $y = 1$

$$\sum_{k=0}^{n-1}x^{n-1-k} = \frac{x^n-1}{x-1}$$

This is just to explain the 'special' factoring you aksed about.

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@Anne_Bauval already gave the limits in their answer, namely

$$\lim_{x \rightarrow 1}\frac{x^n-1}{x^m-1} = \frac{n}{m}$$ and $$\lim_{x \rightarrow 1}\frac{x^n-1 - n(x-1)}{(x-1)^2} = \frac{n(n-1)}{2}$$

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Relation to Combinatorics

This isn't directly related to combinatorics without defining what $x, y, n, m$ are. There are perhaps a few links

1. Is a link to the binomial coefficient $$\lim_{x \rightarrow 1}\frac{x^n-1 - n(x-1)}{(x-1)^2} = \frac{n(n-1)}{2} = {n \choose 2}$$

2. In the field of post quantum calculus one of the key building blocks of the field: the $(p, q)$-number is based off of $x^n-y^n$, and is defined as $$\frac{p^n-q^n}{p-q}$$ which (when using the formula from above) yields $$\frac{p^n-q^n}{p-q} = \sum_{k=0}^{n-1}p^{n-1-k}q^k$$ (although this is called post quantum 'calculus', it is a field of combinatorics in its own right)

3. Binet's formula

If we have the two roots of $x^2-x-1$ (golden ratio) denoted as $\phi_+, \phi_-$ then the nth Fibonacci number $F_n$ is $$F_m = \frac{\phi_+^n-\phi_-^n}{\phi_+-\phi_-}$$

These are just some examples of potential relations to combinatorics