Answer
$\theta=10k+5$, $k$ integer
Work Step by Step
We are given the function:
$g(\theta)=\tan\dfrac{\pi \theta}{10}$
We should compute $\lim\limits_{\theta \to a} g(\theta)=\lim\limits_{\theta \to a}\tan\dfrac{\pi \theta}{10}$
$=\lim\limits_{\theta \to a}\dfrac{\sin\dfrac{\pi \theta}{10}}{\cos\dfrac{\pi \theta}{10}}$
The function $g(\theta)$ is undefined for $\cos\dfrac{\pi \theta}{10}=0$.
$\dfrac{\pi \theta}{10}=\dfrac{(2k+1)\pi}{2}$
$\dfrac{\pi \theta}{10}=\dfrac{5(2k+1)\pi}{10}$
$\theta=10k+5$
Therefore $g(\theta)$ has vertical asymptotes in $\theta=10k+5$, $k$ integer.
