Function Limits of Polynomials

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

Prove that \begin{equation}\lim_{x\to a}\,P_n(x)=P_n(a)\end{equation} where $P_n(x)$ is any arbitrary polynomial of degree $n$.

Answer: 

It is true that \begin{equation}\lim_{x\to a}\,P_n(x)=P_n(a)\end{equation} where $P_n(x)$ is any arbitrary polynomial of degree $n$.

SAMPLE SOLUTION

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Solution: 

We use the notation of <Sign in to see all the formulas> to denote a Polynomial Function. In particular,

<Sign in to see all the formulas>

<Sign in to see all the formulas> is the sum of functions that we denote by

<Sign in to see all the formulas>


We know that the limit of a sum of functions is the sum of the individual limits if they exists. Therefore,

<Sign in to see all the formulas>

We now need to determine if each <Sign in to see all the formulas> exists. Each <Sign in to see all the formulas> is basically a constant times another function. Let's say <Sign in to see all the formulas>. We know that

<Sign in to see all the formulas>


exists if

<Sign in to see all the formulas>


exists.

Because <Sign in to see all the formulas> is in the form $x^n$, we know that

<Sign in to see all the formulas>


so

<Sign in to see all the formulas>


does exists and is equal to $a^n$. Finally, <Sign in to see all the formulas> is a constant and we know that the limit of a constant is the constant itself.

Therefore,

<Sign in to see all the formulas>

Now because

<Sign in to see all the formulas>


a polynomial is a continuous function.

Finally, because we never specified what the value of $a$ was, it is an arbitrary number. Therefore, <Sign in to see all the formulas> is continuous on the entire $x$-axis.