# How do you find the Vertical, Horizontal, and Oblique Asymptote given s(t) = t / sin t?

Jan 20, 2018

Vertical asymptotes where $t = n \pi : n \in \mathbb{Z} , n \ne 0$

#### Explanation:

$s \left(t\right) = \frac{t}{\sin} t$

$s \left(t\right)$ is undefined whereever $\sin t = 0$

I.e Where $t = n \pi : \forall n \in \mathbb{Z}$

Now consider the graph of $s \left(t\right)$ below.

graph{x/sinx [-46.2, 46.33, -23.06, 23.16]}

It can be seen that $s \left(t\right)$ has vertical asymptotes where $t = n \pi : n \in \mathbb{Z} , n \ne 0$

Also note ${\lim}_{t \to o} \frac{t}{\sin} t = 1$ can be seen on the graph above.

$s \left(t\right)$ has no other asymptotes.