Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 13

Show that the function $\displaystyle f(x) = \frac{2x+3}{x-2}$ is continuous on the interval $(2,\infty)$ by using the definition
of continuity and the properties of limits.

By using the properties of limit, let's pick $a=3$ on the interval $(2,\infty)$


$
\begin{equation}
\begin{aligned}
\lim\limits_{x \to 3} \frac{2x+3}{x-2} & = \frac{
2\lim\limits_{x \to 3} x + \lim\limits_{x \to 3} 3
}
{
\lim\limits_{x \to 3} x - \lim\limits_{x \to 3} 2
}
&& \text{(Applying Difference, Sum and Quotient Law.)}\\
& = \frac{2(3)+3}{3-2}
&& \text{(Substitute } a = 3)\\
& = 9
&& \text{(It shows that the function is continuous at 3 and is equal to 9)}
\end{aligned}
\end{equation}
$


By using the definition of continuity,
The given function is a rational function that is continuous at every number in its domain according to the theorem. And the domain of the function is $(-\infty,2) \, \bigcup \, (2, \infty)$

Therefore,
The function is continuous on the interval $(2,\infty)$