A polynomial asymptote is a function

such that


Since this equation defines a hyperbola, we expect the asymptotes to be lines of the form

.
Ignore the negative root (we don't need it). If

, then we want to find constants

such that

We have



since

forces us to have

. And as

, the

term is "negligible", so really

. We can then treat the limand like

which tells us that we would choose

. You might be tempted to think

, but that won't be right, and that has to do with how we wrote off the "negligible" term. To find the actual value of

, we have to solve for it in the following limit.


We write

Now as

, we see this expression approaching

, so that

So one asymptote of the hyperbola is the line

.
The other asymptote is obtained similarly by examining the limit as

.


Reduce the "negligible" term to get

Now we take

, and again we're careful to not pick

.



This time the limit is

, so

which means the other asymptote is the line

.