# Limit of a Polynomial Function

In this tutorial we shall look at the limit of a polynomial function of any degree, and this is useful to solve different polynomial functions’ limits.

If $P\left( x \right) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + {a_{n – 1}}{x^{n – 2}} + \cdots + {a_2}{x^2} + {a_1}x + {a_0}$ is a polynomial function of degree $n$, show that $\mathop {\lim }\limits_{x \to k} P\left( k \right)$

We have the given polynomial function of degree $n$
$P\left( x \right) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + {a_{n – 1}}{x^{n – 2}} + \cdots + {a_2}{x^2} + {a_1}x + {a_0}\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Putting $x = k$ in the above equation (i), we have
$P\left( k \right) = {a_n}{k^n} + {a_{n – 1}}{k^{n – 1}} + {a_{n – 1}}{k^{n – 2}} + \cdots + {a_2}{k^2} + {a_1}k + {a_0}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)$

Taking the limit of equation (i) as $x \to k$, we have
$\mathop {\lim }\limits_{x \to k} P\left( x \right) = \mathop {\lim }\limits_{x \to k} \left( {{a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + {a_{n – 1}}{x^{n – 2}} + \cdots + {a_2}{x^2} + {a_1}x + {a_0}} \right)$

Appling the value of the limit, we get
$\begin{gathered} \mathop {\lim }\limits_{x \to k} P\left( x \right) = \mathop {\lim }\limits_{x \to k} \left( {{a_n}{x^n}} \right) + \mathop {\lim }\limits_{x \to k} \left( {{a_{n – 1}}{x^{n – 1}}} \right) + \mathop {\lim }\limits_{x \to k} \left( {{a_{n – 1}}{x^{n – 2}}} \right) + \cdots + \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to k} \left( {{a_2}{x^2}} \right) + \mathop {\lim }\limits_{x \to k} \left( {{a_1}x} \right) + \mathop {\lim }\limits_{x \to k} \left( {{a_0}} \right) \\ \Rightarrow \mathop {\lim }\limits_{x \to k} P\left( x \right) = {a_n}\mathop {\lim }\limits_{x \to k} \left( {{x^n}} \right) + {a_{n – 1}}\mathop {\lim }\limits_{x \to k} \left( {{x^{n – 1}}} \right) + {a_{n – 1}}\mathop {\lim }\limits_{x \to k} \left( {{x^{n – 2}}} \right) + \cdots + \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a_2}\mathop {\lim }\limits_{x \to k} \left( {{x^2}} \right) + {a_1}\mathop {\lim }\limits_{x \to k} \left( x \right) + {a_0} \\ \Rightarrow \mathop {\lim }\limits_{x \to k} P\left( x \right) = {a_n}{k^n} + {a_{n – 1}}{k^{n – 1}} + {a_{n – 1}}{k^{n – 2}} + \cdots + {a_2}{k^2} + {a_1}k + {a_0} \\ \end{gathered}$

Using equation (ii), we have
$\Rightarrow \mathop {\lim }\limits_{x \to k} P\left( x \right) = P\left( k \right)$