I need to find the asymptotes of a parametric equation. My book says you have a vertical asymptote when $y\to \infty$. But the parametric equation is the following:
$$x= \frac 13t^3-\pi,y= \frac 13t^3 -2\arctan(t)$$
When does $y$ ever go to infinity? I thought about $t=\infty$, but the $\arctan(\infty)$ does not exist does it?
With $$x=\frac{1}{3}t^3 - \pi, \ y = \frac{1}{3}t^3 - 2 \arctan t,$$ you have $$ y= x + \pi -2 \arctan t. $$ Since $\arctan t$ goes to $\frac{\pi}{2}$ as $t \to \infty$ and $-\frac{\pi}{2}$ as $t \to -\infty$, your two asymptotes are $$ y = x + \pi - 2\left(\frac{\pi}{2}\right) = x$$ and $$ y = x +\pi - 2 \left(-\frac{\pi}{2}\right) = x+2\pi.$$