While calculating the limit of quadratic functions, the $$\lim_{x \to a}\frac{p(x)}{q(x)} = \frac{(x - a)\cdot something}{(x-a) \cdot something}$$ formula is commonly used (well, I know stating it like this is probably not particularly precise, but I hope you get the idea). For example: $$\lim_{x \to 4}\frac{x^2 -9x + 20}{x^2 - x -12} = \lim_{x \to 4}\frac{(x-4)(x-5)}{(x-4)(x+3)}$$
My question is: what is the name of this formula and why does it work?
Thank you!
If it has a name, I don't know it. But, yes, it is true that if $a$ is a root of both polynomials $p(x)$ and $q(x)$, that is, if can write $p(x)$ as $(x-a)p^\star(x)$ and $q(x)=(x-a)q^\star(x)$, where $p^\star(x)$ and $q^\star(x)$ are also poynomials, then, indeed,$$\lim_{x\to a}\frac{p(x)}{q(x)}=\lim_{x\to a}\frac{p^\star(x)}{q^\star(x)},$$assuming that the later limit exists. This is so because\begin{align}\require{cancel}\lim_{x\to a}\frac{p(x)}{q(x)}&=\lim_{x\to a}\frac{\cancel{(x-a)}p^\star(x)}{\cancel{(x-a)}q^\star(x)}\\&=\lim_{x\to a}\frac{p^\star(x)}{q^\star(x)}.\end{align}