## Precalculus (6th Edition) Blitzer

The statement “f and g are both continuous at $a$, although $\frac{f}{g}$ is not” makes sense.
For a function to be continuous at a point a, the function must satisfy the following three conditions: (a) f is defined at a. (b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists. (c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ Since, f and g are continuous functions at a, both of them satisfy the above three conditions. Now check whether the function $\frac{f}{g}$ is continuous at a, or not. Find the value of $\left( \frac{f}{g} \right)\left( x \right)$ at $a$, $\left( \frac{f}{g} \right)\left( a \right)=\frac{f\left( a \right)}{g\left( a \right)}$ If $g\left( a \right)=0$, then the value $\left( \frac{f}{g} \right)\left( a \right)$ is not defined. In this case, the function $\left( \frac{f}{g} \right)\left( x \right)$ does not satisfy all the conditions of being continuous. Thus, the function $\left( \frac{f}{g} \right)\left( x \right)$ is not continuous at a. Hence, the statement makes sense.