Answer
please see step-by-step
Work Step by Step
$p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$
$\displaystyle \lim_{x\rightarrow a}p(x)=\lim_{x\rightarrow a}(a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n})=$
... Law 1, The limit of a sum...
=$\displaystyle \lim_{x\rightarrow a}a_{0}+\lim_{x\rightarrow a}a_{1}x+\lim_{x\rightarrow a}a_{2}x^{2}+\cdots+\lim_{x\rightarrow a}a_{n}x^{n}$
... Law 3, The limit of a constant times a function...
=$\displaystyle \lim_{x\rightarrow a}a_{0}+a_{1}\lim_{x\rightarrow a}x+a_{2}\lim_{x\rightarrow a}x^{2}+\cdots+a_{n}\lim_{x\rightarrow a}x^{n}$
... Law 7. $\displaystyle \lim_{x\rightarrow a}c=c$, ...Law 9. $\displaystyle \lim_{x\rightarrow a}x^{n}=a^{n}$,
$=a_{0}+a_{1}a+a_{2}a^{2}+\cdots+a_{n}a^{n}=p(a)$
So,
$\displaystyle \lim_{x\rightarrow a}p(x)=p(a)$
