Answer
The limit of a polynomial function $ f\left( x \right)={{b}_{n}}{{x}^{n}}+{{b}_{n-1}}{{x}^{n-1}}+\cdots +{{b}_{1}}x+{{b}_{0}}$ is $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$, which is the value of the polynomial evaluated at the $ a $.
Work Step by Step
A polynomial function is a sum of monomials.
$ f\left( x \right)={{b}_{n}}{{x}^{n}}+{{b}_{n-1}}{{x}^{n-1}}+\cdots +{{b}_{1}}x+{{b}_{0}}$
Thus, to find the limit of a polynomial, find the limit of each of the monomials and add them up.
So,
$\begin{align}
& \underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,\left( {{b}_{n}}{{x}^{n}}+{{b}_{n-1}}{{x}^{n-1}}+\cdots +{{b}_{1}}x+{{b}_{0}} \right) \\
& =\underset{x\to a}{\mathop{\lim }}\,{{b}_{n}}{{x}^{n}}+\underset{x\to a}{\mathop{\lim }}\,{{b}_{n-1}}{{x}^{n-1}}+\cdots +\underset{x\to a}{\mathop{\lim }}\,{{b}_{1}}x+\underset{x\to a}{\mathop{\lim }}\,{{b}_{0}}
\end{align}$
Use $\underset{x\to a}{\mathop{\lim }}\,c=c\ \text{with }c={{b}_{0}}$,
$\begin{align}
& \underset{x\to a}{\mathop{\lim }}\,f\left( x \right)={{b}_{n}}{{a}^{n}}+{{b}_{n-1}}{{a}^{n-1}}+\cdots +{{b}_{1}}a+{{b}_{0}} \\
& =f\left( a \right)
\end{align}$
which is equal to value of the polynomial evaluated at $ a $.
Thus, if $ f $ is a polynomial function, then $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ for any number $ a $.
For example, $ f\left( x \right)={{x}^{3}}-2{{x}^{2}}+x+3$
$\begin{align}
& \underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to 0}{\mathop{\lim }}\,\left( {{x}^{3}}-2{{x}^{2}}+x+3 \right) \\
& =\underset{x\to 0}{\mathop{\lim }}\,{{x}^{3}}-\underset{x\to 0}{\mathop{\lim }}\,2{{x}^{2}}+\underset{x\to 0}{\mathop{\lim }}\,x+\underset{x\to 0}{\mathop{\lim }}\,3 \\
& ={{0}^{3}}-2\times {{0}^{2}}+0+3 \\
& =0-0+0+3 \\
& =3
\end{align}$
