Vertical asymptote:
Find the restriction on x. This is the easiest of the three asymptotes you will need to find (even if only two can show at a time). As a hyperbola is in the form:
![\frac{1}{x}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bx%7D)
, you only need to find the restriction on the denominator, namely denominator can never be zero. Hence, let the denominator equal to zero to find the vertical asymptote.
Horizontal asymptote:
Find the restriction on y. To do this, you need to simplify the top and bottom to its lowest terms. If it simplifies to a form such as:
![a + \frac{b}{x}](https://tex.z-dn.net/?f=a%20%2B%20%5Cfrac%7Bb%7D%7Bx%7D)
, then the horizontal asymptote becomes y = a. You need to think to yourself, as x grows to infinite, and shrinks to negative infinite, what happens to the function? Does it slowly curve to a stop?
Oblique asymptote:
This is a pretty rare kind, but it still exists, so don't be naive to this sort of asymptote. This is a form of horizontal and vertical asymptote, only it's at an angle. That is, this asymptote is a set of x and y-coordinates that work in unison to produce a curvature or line.
Let's consider:
![f(x) = \frac{x^{2} - 6x + 7}{x + 5}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%5E%7B2%7D%20-%206x%20%2B%207%7D%7Bx%20%2B%205%7D)
Now, in normal term, a horizontal asymptote would have a degree higher in the denominator than in the numerator. However, it's flipped in this case.
Now, you will need to long divide this set of polynomials to yield a straight y = x line, except it's been moved 11 units to the right to yield a y = x - 11 line.
Remember: these exist because the highest power is in the numerator and not the denominator.