Chapter 14.2, Problem 4AYU

To determine

To find: If the statement “the limit of a polynomial function as x approaches 5 equals the value of the polynomial at 5” is true of false..

Answer to Problem 4AYU

The statement is true.

Explanation of Solution

Calculation:

Consider a polynomial function $f\left(x\right)=a{x}^2+bx+c$ . Then, $\underset{x\to 5}{lim}f\left(x\right)$ is calculated as follows:

  $\begin{array}{c}\underset{x\to 5}{lim}f\left(x\right)=\underset{x\to 5}{lim}\left(a{x}^2+bx+c\right)\\ =\underset{x\to 5}{lim}a{x}^2+\underset{x\to 5}{lim}bx+\underset{x\to 5}{lim}c\\ =a{\left(5\right)}^2+b\left(5\right)+c\\ =f\left(5\right)\end{array}$

So, the limit of the polynomial function $f\left(x\right)$ as x approaches 5 is equal to $f\left(5\right)$ , that is the value of the function at 5.

Hence, the given statement is true.