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 

.