Limit of Rational Functions: $p(x)/q(x)$ as $x\to\infty$

[Jameson]

Thank your to Chris L T521 for submitting this week's high school level problem!

$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ and $q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots+b_1x+b_0$. Show that
\[\lim_{x\to\infty}\frac{p(x)}{q(x)}=\begin{cases} \infty & \text{ if $n>m$}\\ \frac{a_n}{b_m} & \text{ if $n=m$}\\ 0 & \text{ if $n<m$}\end{cases}\]

 

[Jameson]

Congratulations to the following members for their correct solutions:

1) Sudharaka
2) Siron

Honorable mention to veronica1999 for a good intuitive explanation but not quite formal enough to constitute a proof.

Solution (from Sudharaka):

[sp]\[p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\mbox{ and }q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots+b_1x+b_0\]

\[\Rightarrow\frac{p(x)}{q(x)}=\frac{x^n}{x^m}\left(\frac{a_n+a_{n-1}x^{-1}+\cdots+a_1x^{-n+1}+a_0x^{-n}}{b_m+b_{m-1}x^{-1}+\cdots+b_1x^{-m+1}+b_0x^{-m}}\right)\]

\(\mbox{Note that, }\displaystyle\lim_{x\rightarrow\infty}\left(\frac{a_n+a_{n-1}x^{-1}+\cdots+a_1x^{-n+1}+a_0x^{-n}}{b_m+b_{m-1}x^{-1}+\cdots+b_1x^{-m+1}+b_0x^{-m}}\right)=\frac{a_n}{b_n}\)

\[\therefore\lim_{x\rightarrow\infty}\frac{p(x)}{q(x)}=\frac{a_n}{b_n}\lim_{x\rightarrow\infty}x^{n-m}\]

\(\mbox{Note that, }\displaystyle\lim_{x\rightarrow\infty}x^{n-m}=\begin{cases} \infty & \text{ if $n>m$}\\ 1 & \text{ if $n=m$}\\ 0 & \text{ if $n<m$}\end{cases}\)

\[\therefore\lim_{x\rightarrow\infty}\frac{p(x)}{q(x)}=\begin{cases} \infty & \text{ if $n>m$}\\ \frac{a_n}{b_m} & \text{ if $n=m$}\\ 0 & \text{ if $n<m$}\end{cases}\]

Q.E.D [/sp]