Vertical and Horizontal Asymptotes

A horizontal asymptote is an imaginary line drawn across a coordinate plane that acts as a limit to a function.  In other words, if there were a horizontal asymptote at y = 3, as x moved forward the graph would continue to approach y = 3 but would never actually reach 3.  Here is an example from www.wikipedia.org that illustrates my point.  

Notice how the lines of the graph approach the horizontal dotted lines (the asymptotes) but never reach them.

A vertical asymptote is very similar, but instead of being horizontal lines that limit the graph, they are vertical lines that help to give shape to the graphical representation of the function.  Here is another picture (also from www.wikipedia.org) to better explain what I mean.  

In this case, the y-axis acts as the asymptote.

Finding the asymptotes is relatively simple.

For a horizontal asymptote, we write the equation in standard form such as  and we ignore everything except the numbers with the highest powers (the x^2 and the 4x^2) and divide them as normal.  The x^2 cancels out, and we are left with our horizontal asymptote, 1/4.

For vertical asymptotes, we again write the equation in standard form  and this time just find the zeroes of the denominator.  Just set the denominator to zero and solve the resulting equation.  

4x^2 - 9 = 0

4x^2 = 9

x^2 = 9/4

x = plus or minus 3/2

So in this case, we have a vertical asymptote at x = 3/2, another at - 3/2, and a horizontal asymptote at y = 1/4.

That's how it's done.