Given the domain $\displaystyle 0 \leq x \leq 2\pi$ in $\displaystyle F(x) = x \cdot tan(x) - \frac{sin(x)}{x}$, is it possible to for $\displaystyle tan(x)$ to cause $\displaystyle F$ to be undefined for $\displaystyle x = \frac{\pi}{2} \pm 2\pi K$, where $\displaystyle K$ is a natural number? Given the domain restriction, it would seem that ONLY $\displaystyle x = \{\frac{\pi}{2}, \frac{3\pi}{2}\}$ in $\displaystyle tan(x)$ would cause $\displaystyle F$ to be undefined. Divide by zero is trivially obvious and is not in question.

The solution key says that $\displaystyle x$ in $\displaystyle tan(x)$ will cause $\displaystyle F$ to be undefined for all $\displaystyle x = \frac{\pi}{2} \pm 2\pi K$.

$\displaystyle \frac{\pi}{2} + 2\pi 1$ is $\displaystyle \frac{5\pi}{2}$, which is greater than the $\displaystyle 2\pi$ limit of $\displaystyle x$, and $\displaystyle \frac{\pi}{2} - 2\pi 1$ is $\displaystyle - \frac{3\pi}{2}$, which is less than the $\displaystyle 0$ limit of $\displaystyle x$. Their solution would also exclude $\displaystyle x = \frac{3\pi}{2}$. Furthermore, the only value of $\displaystyle K$ that will keep $\displaystyle x$ within the domain restriction is $\displaystyle 0$. So, why do they bring "$\displaystyle \pm 2\pi K$" into consideration at all?