Chapter 1.3, Problem 42ES

In each part, find examples of polynomials $p\left(x\right)$ and $q\left(x\right)$ that satisfy the stated condition and such that $p\left(x\right)\to +\infty$ and $q\left(x\right)\to +\infty$ as $x\to +\infty$ .

(a) $\underset{x\to +\infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=1$

(b) $\underset{x\to +\infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=0$

(c) $\underset{x\to +\infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=+\infty$

(d) $\underset{x\to +\infty }{\lim }\left[p\left(x\right)-q\left(x\right)\right]=3$