Horizontal Asymptote of a Polar Equation

If you graph the polar function $\displaystyle r = 1 - cot(\theta)$, $\displaystyle 0\leq\theta\leq\2\pi$, $\displaystyle -10\leq x \leq\10$, $\displaystyle -2\leq y \leq\2$, you should find a horizontal asymptote. Prove there is one, and find the other, not visible horizontal asymptote

I tried converting it back to cartesian but the equation cannot be isolated for y so that I could take the limit as x approaches positive or negative infinity. Other than that, I have no idea how to approach this problem. Any help would be appreciated.

Re: Horizontal Asymptote of a Polar Equation

Quote:

Originally Posted by

**freestar** If you graph the polar function $\displaystyle r = 1 - cot(\theta)$, $\displaystyle 0\leq\theta\leq\2\pi$, $\displaystyle -10\leq x \leq\10$, $\displaystyle -2\leq y \leq\2$, you should find a horizontal asymptote. Prove there is one, and find the other, not visible horizontal asymptote

I tried converting it back to cartesian but the equation cannot be isolated for y so that I could take the limit as x approaches positive or negative infinity. Other than that, I have no idea how to approach this problem. Any help would be appreciated.

Hi freestar! :)

You would get an asymptote if r tends to infinity.

At which values of $\displaystyle \theta$ does r tend to infinity?

Re: Horizontal Asymptote of a Polar Equation

I tried to find that limit but I can't seem to find it because plugging in large numbers does not yeild one specific number. Confused. Sorry if I am being dumb.

Thanks a lot

Re: Horizontal Asymptote of a Polar Equation

At which values of θ is cot(θ) undefined?

Re: Horizontal Asymptote of a Polar Equation

Re: Horizontal Asymptote of a Polar Equation

Quote:

Originally Posted by

**freestar** 0, pi, 2pi, 3pi, etc..

Right!

Let's stick to the ones within your domain, which are 0 and pi.

What are the x and y coordinates that correspond to those angles (or angles close to them)?

Re: Horizontal Asymptote of a Polar Equation

Ahh!! I see that y = $\displaystyle \pm1$ are horizontal asymptotes but what about the ones that is not visible. Still unclear on what that means.

Re: Horizontal Asymptote of a Polar Equation

You can approach 0 and pi both from different sides.

This leads to 4 asymptotic relations.

In 2 of those r tends to +infinity.

In the other 2, r tends to -infinity.

I can only assume that the last 2 are considered "not visible", since in polar coordinates r should be positive.

Re: Horizontal Asymptote of a Polar Equation