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
.
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
We have
since
which tells us that we would choose
We write
Now as
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
This time the limit is
which means the other asymptote is the line