Chapter 14.3, Problem 70AYU

To determine

To find: The numbers at which $f$ is continuous.

Answer to Problem 70AYU

Continuous for all real values of $x$ excluding $x=\pm 3$.

Explanation of Solution

Given:

$f\left(x\right)=\frac{x^2-4}{x^2-9}$

Calculation:

$f\left(x\right)=\frac{x^2-4}{x^2-9}$ is a rational function.

The denominator of $f\left(x\right)$ becomes zero for $x^2-9=0$ ie $x=\pm 3$.

Since $f\left(x\right)=\frac{x^2-4}{x^2-9}=\frac{\left(x+2\right)\left(x-2\right)}{\left(x+3\right)\left(x-3\right)}$ is undefined for $x=\pm 3$; therefore $f\left(x\right)=\frac{x^2-4}{x^2-9}$ is continuous for all real numbers excluding $x=\pm 3$.