Chapter 14.2, Problem 5AYU

To determine

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

Answer to Problem 5AYU

The statement is false as explained below.

Explanation of Solution

Calculation:

Consider a rational function $f\left(x\right)=\frac{x^2-6x+5}{x^2-25}$ . The function is not defined at $x=5$ as the denominator becomes 0 for $x=5$ .

The $\underset{x\to 5}{lim}f\left(x\right)$ is calculated as follows:

  $\begin{array}{c}\underset{x\to 5}{lim}f\left(x\right)=\underset{x\to 5}{lim}\left(\frac{x^2-6x+5}{x^2-25}\right)\\ =\underset{x\to 5}{lim}\left(\frac{2x-6}{2x}\right)\left[ByL\text{'}HopitalRule\right]\\ =\frac{2\left(5\right)-6}{2\left(5\right)}\\ =\frac{2}{5}\end{array}$

So, it can be concluded that the limit of arational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ at 5 is equal to the value of the function at 5, only if $q\left(x\right)\ne 0$ at 5.

Hence, the given statement is false.