(related to Proposition: Limit of a Rational Function)

- Let $p$, $q$ be any polynomials.

- Let $a$ be a real number with $q(a)\neq 0.$

- Since $p$ and $q$ are polynomials, it follows from the limit of polynomials that $\lim_{x\to a}p(x)=p(a),$ and $\lim_{x\to a}q(x)=q(a).$

- From the limit of a quotient it follows for $q(a)\neq 0$ and the
rational function $\frac{p(x)}{q(x)}$ that $$\lim_{x\to a}\frac{p(x)}{q(x)}=\frac{p(a)}{q(a)}.$$∎

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016