# Question 78ac4

Nov 29, 2016

The asymptotes are in x = pi/6+kpi/3; k in ZZ

#### Explanation:

To simplify let's put:

$t = 3 x$

and first analyse $f \left(t\right) = \tan \frac{t}{t}$.

The vertical asymptotes of $\tan t$ are located in $t = \frac{\pi}{2} + k \pi$ for $k \in \mathbb{Z}$.

if $t \ne 0$ this obviously does not change.

For $t = 0$ we have:

${\lim}_{t \to 0} \tan \frac{t}{t} = {\lim}_{t \to 0} \sin \frac{t}{t} \cdot \frac{1}{\cos} t = 1$

As $t = 3 x$, the vertical asymptotes of $f \left(x\right) = \tan \frac{3 x}{3 x}$ are located in:

x = pi/6+kpi/3; k in ZZ#

graph{tan(3x)/(3x) [-5, 5, -2.5, 2.5]}