<s>[INST] How is $\lim_{x \to a}\left(\frac{x^n - a^n}{x - a}\right) = n\times a^{n-1}$? In my book this is termed as a theorem and the proof given is as follows :-
$$\begin{align}
\lim_{x \to a}\left(\frac{x^n - a^n}{x - a}\right)
&=\lim_{x \to a}\left(\frac{(x - a)*(x^{x-1} + x^{n-2}*a + x^{n-3}*a^2 + x^{n-4}*a^3 + \cdots + x^1*a^{n-2} + a^{n-1})}{x - a}\right) \\ &=(a^{n-1} + a*a^{n-2} + \cdots + a^{n-1}) \\ &=(a^{n-1} + a^{n-1} + \cdots + a^{n-1}) \\ &=(n*a^{n-1}).
\end{align}$$
Everything made sense to me except 
$$x^n - a^n = (x - a)*(x^{x-1} + x^{n-2}*a + x^{n-3}*a^2 + x^{n-4}*a^3 + \cdots + x^1*a^{n-2} + a^{n-1})$$
Somebody please enlighten me on this topic. 
 [/INST] No, this isn't a binomial expansion. This is just long division. To check that it makes sense, we can just expand it out:
\begin{align*}
&(x - a)(x^{n - 1} + ax^{n - 2} + a^2x^{n - 3} + \cdots + a^{n - 2}x + a^{n - 1}) \\
&= x(x^{n - 1} + ax^{n - 2} + a^2x^{n - 3} + \cdots + a^{n - 1}) \\
&~~~~~~~~~~~~~~~~~- a(x^{n - 1} + ax^{n - 2} + \cdots + a^{n - 2}x + a^{n - 1}) \\
&= (x^n \color{red}{+ ax^{n - 1} + a^2x^{n - 2} + \cdots  + a^{n - 1}x}) \\
&~~~~~~~+ (\color{red}{-ax^{n - 1} - a^2x^{n - 2} - \cdots - a^{n - 1}x} - a^n) \\
&= x^n - a^n
\end{align*}
</s>