Let P and Q be polynomials and consider the limit of P(x)/Q(x) as x approaches infinity. State and prove a conjecture for the value of this limit that depends on the degrees of P and Q and their leading coefficients.
Let $\displaystyle P=p_n x^n+p_{n-1}x^{n-1}+...+p_1 x+p_0$ and let $\displaystyle Q=q_m x^m+q_{m-1}x^{m-1}+...+q_1 x+q_0$.
If $\displaystyle deg(P)>deg(Q)$, then $\displaystyle \lim_{x\to\infty}\frac{P(x)}{Q(x)}=\infty$.
If $\displaystyle deg(P)<deg(Q)$, then $\displaystyle \lim_{x\to\infty}\frac{P(x)}{Q(x)}=0$.
If $\displaystyle deg(P)=deg(Q)$, then $\displaystyle \lim_{x\to\infty}\frac{P(x)}{Q(x)}=\frac{p_n}{q_m}$.
The proofs are fairly straightforward. If $\displaystyle k=\min\{n,m\}$, just factor out $\displaystyle x^k$ from both polynomials and see what you have left.