# Why can't a graph...

• Feb 5th 2008, 04:58 PM
Why can't a graph...
...have more than two horizontal asymptotes?
I just want to know for sure.
• Feb 5th 2008, 05:30 PM
mr fantastic
Quote:

...have more than two horizontal asymptotes?
I think its because the graph can only have an asymptote as one of its boundaries, and a graph can only be bounded on two sides. I just want to know for sure.

You find horizontal asymptotes by considering the limiting behavior of y = f(x) as x approaches +oo and -oo:

If $\displaystyle \lim_{x \rightarrow -\infty} f(x) = a$ then $\displaystyle y \rightarrow a$ and the line $\displaystyle y = a$ is a horizontal asymptote.

If $\displaystyle \lim_{x \rightarrow +\infty} f(x) = b$ then $\displaystyle y \rightarrow b$ and the line $\displaystyle y = b$ is a horizontal asymptote.

When they exist, the values of a and b are unique by the 'uniquesness of a limit theorem'.

Therefore there are no more than two possible horizontal asymptotes.

Stick at it (Rofl)