Limit of a Polynomial Function

In this tutorial we shall give idea of limit of a polynomial function of any degree which 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 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 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)

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