## Precalculus (6th Edition) Blitzer

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 $a$.
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}