Let $f(x) = \sin{x} + 2\cos{x} + 3\tan{x}$, using radian measure for the variable $x$. In what interval does the smallest positive value of $x$ for which $f(x) = 0$ lie?
$\textbf{(A)}\ (0,1)  \qquad \textbf{(B)}\ (1, 2) \qquad\textbf{(C)}\ (2, 3) \qquad\textbf{(D)}\ (3, 4) \qquad\textbf{(E)}\ (4,5)$
We must first get an idea of what $f(x)$ looks like:
Between $0$ and $1$, $f(x)$ starts at $2$ and increases; clearly there is no zero here.
Between $1$ and $\frac{\pi}{2}$, $f(x)$ starts at a positive number and increases to $\infty$; there is no zero here either.
Between $\frac{\pi}{2}$ and 3, $f(x)$ starts at $-\infty$ and increases to some negative number; there is no zero here either.
Between $3$ and $\pi$, $f(x)$ starts at some negative number and increases to -2; there is no zero here either.
Between $\pi$ and $\pi+\frac{\pi}{4} 0$. There is a zero here by the Intermediate Value Theorem. Therefore, the answer is $\boxed{\textbf{(D)}}$.