Some functions are continuous from negative infinity to positive infinity, but others break off at a point of discontinuity or turn off and never make it past a certain point. Vertical and horizontal asymptotes are straight lines that define the value the function approaches if it does not extend to infinity in opposite directions. Horizontal asymptotes are always in the form y = C, and vertical asymptotes are always in the form x = C, where C is any constant. Both horizontal and vertical asymptotes are the easy to find.
Write the function for which you are trying to find a vertical asymptote. These most likely will be rational functions, with the variable x somewhere in the denominator. When the denominator of a rational function approaches zero, it has a vertical asymptote.
Find the value of x that makes the denominator equal to zero. If your function is y = 1/(x+2), you would solve the equation x+2 = 0, which is x = -2. There may be more than one possible solution for more complex functions.
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Take the limit of the function as x approaches the value you found from both directions. For this example, as x approaches -2 from the left, y approaches negative infinity; when -2 is approached from the right, y approaches positive infinity. This means the graph of the function splits at the discontinuity, jumping from negative infinity to positive infinity. Do this for each value individually if multiple solutions were found in the previous step.
Write the equations of the asymptotes by setting x equal to each of the values used in the limits. For this example, there is only one asymptote, which is given by the equation x = -2.
Write your function. Horizontal asymptotes can be found in a wide variety of functions. For this example, the function is y = x/(x-1).
Take the limit of the function as x approaches infinity. In this example, the "1" can be ignored because it becomes insignificant as x approaches infinity. Infinity minus 1 is still infinity. So, the function becomes x/x, which equals 1. Therefore, the limit as x approaches infinity of x/(x-1) = 1.
Use the solution of the limit to write your asymptote equation. If the solution is a fixed value, there is a horizontal asymptote, but if the solution is infinity, there is no horizontal asymptote. If the solution is another function, there is an asymptote, but it is neither horizontal or vertical. For this example, the horizontal asymptote is y = 1.
Trigonometric functions that have asymptotes can be solved in the same way, using the various limits. Realize that trig functions are cyclical and may have many asymptotes.