◀ ▲ ▶Branches / Analysis / Proof

Proof

(related to Proposition: Limit of a Rational Function)

Context

• Let $p$, $q$ be any polynomials.

Hypothesis

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

Implications

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

Conclusion

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

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016