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)