Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 21

Explain using theorems of continuity why the function $\displaystyle F(x) = \frac{x}{x^2 + 5x + 6}$ is continuous at every number in its domain. State the domain.

The function $F(x)$ is a rational function that is continuous on its domain according to the definition. The function is defined for all values of $x$ except for the value that will make denominator equal to 0. So,

$x^2 + 5x + 6 = 0$

Using Quadratic Formula,


$
\begin{equation}
\begin{aligned}
x_{(1,2)} & = \frac{-b \pm \sqrt{b^2 - 4 ac }}{2a}\\
x_{(1,2)} & = \frac{-5 \pm \sqrt{(5)^2 - 4(1)(6)}}{2(1)}
\end{aligned}
\end{equation}
$


$x = -2 \quad \text{ and } \quad x = -3$

Therefore,
Domain: $(-\infty, -3) \bigcup (-3, -2) \bigcup (-2, \infty)$

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