Answer
(a) 0
(b) does not exist (either $\infty$ or $-\infty$)
Work Step by Step
Let \[P(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\]
and
\[Q(x)=b_0+b_1x+b_2x^2+\cdots+b_mx^m\]
Let \[L=\lim_{x\rightarrow\infty}\frac{P(x)}{Q(x)}\]
(a) Let $deg P(x)deg Q(x)\;\;\;,\;\;\; $i.e., $n>m$
\[L=\lim_{x\rightarrow\infty}\frac{P(x)}{Q(x)}\]
\[\Rightarrow L=\lim_{x\rightarrow\infty}\frac{a_0+a_1x+a_2x^2+\cdots+a_mx^m+a_{m+1}x^{m+1}+\cdots+a_nx^n}{b_0+b_1x+b_2x^2+\cdots+b_mx^m}\]
Since $n>m$
Divide numerator and denominator by $x^m$
\[\Rightarrow L=\lim_{x\rightarrow\infty}\frac{\displaystyle\frac{a_0}{x^m}+\displaystyle\frac{a_1}{x^{m-1}}+\displaystyle\frac{a_2}{x^{m-1}}+\cdots+\displaystyle a_m+a_{m+1}x+\cdots+a_nx^{n-m}}{\displaystyle\frac{b_0}{x^m}+\frac{b_1}{x^{m-1}}+\frac{b_2}{x^{m-2}}+\cdots+b_m}\]
\[L=\frac{0+0+\cdots+0+a_m+\infty+\infty+\cdots\infty}{0+0+\cdots+0+b_m}\]
\[\Rightarrow L=\infty\]
