Prove the following for a rational function $r(x)=\frac{p(x) }{
q(x)}$ where $p(x)$ and $q(x)$ are polynomials.
- If $deg(p(x))>deg(q(x))$ then $\lim_{x \to \infty}r(x)= \infty (or -
\infty)$
- If $deg(q(x))>deg(p(x))$ then $\lim_{x \to \infty}r(x)= 0$
- If $deg(p(x))=deg(q(x))$ then $\lim_{x \to \infty}r(x)=\frac{\text
{leading coefficient of } p(x)}{ \text {leading coefficient of } q(x)}$
Tags: limits, limits-at-infinity, asymptotes, limits-unbounded, rational, proof, t2, c2
Many students believe that working out a specific example is a proof, and it will take some work to convince them otherwise. Have them work out a specific example in each case, and then try to generalize it.
-- Concha Gomez 1999-02-19