# Question #342c3

Aug 21, 2017

#### Explanation:

$y = 2 \tan \left(4 x - 20\right) = \frac{2 \sin \left(4 x - 20\right)}{\cos} \left(4 x - 20\right)$

has no horizontal asymptottes. (There is no limit as $x \rightarrow \pm \infty$.),
and has vertical asymptotes where the denominator is $0$.

If we are working in degrees

$\cos \left(4 x - {20}^{\circ}\right) = 0$
$4 x - {20}^{\circ} = {90}^{\circ} + k {180}^{\circ}$ for integer $k$
$4 x = {110}^{\circ} + k {180}^{\circ}$ for integer $k$
$x = {22.5}^{\circ} + k {45}^{\circ}$ for integer $k$

The graph has vertical asymptotes at $x = {22.5}^{\circ} + k {45}^{\circ}$ for integer $k$

If we are working in radians or in real numbers

$\cos \left(4 x - 20\right) = 0$
$4 x - 20 = {\pi}^{\circ} + k 2 {\pi}^{\circ}$ for integer $k$
$4 x = \left(\pi + 20\right) + k 2 \pi$ for integer $k$
$x = \frac{\pi}{4} + 5 + + k \frac{\pi}{4}$ for integer $k$

The graph has vertical asymptotes at $x = \frac{\pi}{4} + 5 + + k \frac{\pi}{4}$ for integer $k$