Answer
(a) $\lim\limits_{x \to \infty}\frac{P(x)}{Q(x)} = 0$
(b) $\lim\limits_{x \to \infty}\frac{P(x)}{Q(x)} = \pm \infty$
Work Step by Step
(a) Suppose that the degree of P is less than the degree of Q. Let $n$ be the degree of Q. Let $a_n$ be the coefficient of the term $a_n~x^n$ in the polynomial Q.
We can find the limit:
$\lim\limits_{x \to \infty}\frac{P(x)}{Q(x)} = \frac{\lim\limits_{x \to \infty}P(x)/x^n}{\lim\limits_{x \to \infty} Q(x)/x^n} = \frac{0}{a_n} = 0$
(b) Suppose that the degree of P is greater than the degree of Q. Let $n$ be the degree of Q. Let $a_n$ be the coefficient of the term $a_n~x^n$ in the polynomial Q.
We can find the limit:
$\lim\limits_{x \to \infty}\frac{P(x)}{Q(x)} = \frac{\lim\limits_{x \to \infty}P(x)/x^n}{\lim\limits_{x \to \infty} Q(x)/x^n} = \frac{\pm \infty}{a_n} =\pm \infty$
