Find the Asymptotes y=tan(3/4x)

Combine ${\displaystyle \frac{3}{4}}$ and ${\displaystyle x}$.

For any ${\displaystyle y=\tan \left(x\right)}$, vertical asymptotes occur at ${\displaystyle x=\frac{\pi }{2}+n\pi }$, where ${\displaystyle n}$ is an integer. Use the basic period for ${\displaystyle y=\tan \left(x\right)}$, ${\displaystyle (-\frac{\pi }{2},\frac{\pi }{2})}$, to find the vertical asymptotes for ${\displaystyle y=\tan \left(\frac{3x}{4}\right)}$. Set the inside of the tangent function, ${\displaystyle bx+c}$, for ${\displaystyle y=a\tan (bx+c)+d}$ equal to ${\displaystyle -\frac{\pi }{2}}$ to find where the vertical asymptote occurs for ${\displaystyle y=\tan \left(\frac{3x}{4}\right)}$.

Set the inside of the tangent function ${\displaystyle \frac{3x}{4}}$ equal to ${\displaystyle \frac{\pi }{2}}$.

The basic period for ${\displaystyle y=\tan \left(\frac{3x}{4}\right)}$ will occur at ${\displaystyle (-\frac{2\pi }{3},\frac{2\pi }{3})}$, where ${\displaystyle -\frac{2\pi }{3}}$ and ${\displaystyle \frac{2\pi }{3}}$ are vertical asymptotes.

The vertical asymptotes for ${\displaystyle y=\tan \left(\frac{3x}{4}\right)}$ occur at ${\displaystyle -\frac{2\pi }{3}}$, ${\displaystyle \frac{2\pi }{3}}$, and every ${\displaystyle \frac{4\pi n}{3}}$, where ${\displaystyle n}$ is an integer.

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: ${\displaystyle x=\frac{2\pi }{3}+\frac{4\pi n}{3}}$ where ${\displaystyle n}$ is an integer