Further from the center of the hyperbola, the branches approach, but never cross, two lines known as asymptotes. For a hyperbola with horizontal major axis, the general equation for these asymptotes is: ,
whereas if the major axis is vertical, the general equation for these asymptotes is: .
To prove that these equations represent the asymptotes of a hyperbola, we can look at the most basic case - a hyperbola with horizontal major axis centered at with semi-major axis length a and semi-minor axis length b - and can then extend this result to hyperbolae centered away from the origin.
We must prove that the line is an asymptote of the hyperbola in the first quadrant (all other quadrants follow by symmetry):
So, is the equation of the hyperbola in the first quadrant.
Now, if is an asymptote of the hyperbola as we claim it to be, this should be the line that the hyperbola approaches as x grows infinitely large.
We can take the limit of the difference between these functions as , and if this limit equals 0, then we can be certain that this line is an asymptote.
Therefore the branch of the hyperbola, approaches as in the first quadrant.