Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 26

Explain using theorems of continuity why the function $h(x) = \tan 2x$ is continuous at every number in its domain. State the domain

We can rewrite,

$
\begin{equation}
\begin{aligned}
h(x) =& \tan 2x \quad \text{ as } \quad h(x) = \frac{\sin 2x}{\cos 2x}
\end{aligned}
\end{equation}
$


$h(x)$ is a rational function that is continuous on every number on its domain where $\cos 2x \neq 0$ based from the definition.


$
\begin{equation}
\begin{aligned}
\cos 2x & = 0\\
2x & = \cos^{-1} [0]\\
2x & = \frac{\pi}{2} + n \pi; \quad \text{ Where } n \text{ is an integer and } n \pi \text{ for its multiple cycles.}\\
x & = \frac{\pi}{4} + \frac{n \pi}{2}
\end{aligned}
\end{equation}
$


Therefore,
$\quad$ The domain of $y = \tan 2x$ are all the values of $x$ except for $\displaystyle\frac{x}{4} + \frac{n\pi}{2}$; where $n$ is an integer.