When expressed on a graph, some functions are continuous from negative infinity to positive infinity. However, this is not always the case: other functions break off at a point of discontinuity, or turn off and never make it past a certain point on the graph. Vertical and horizontal asymptotes are straight lines that define the value that a given function approaches if it does not extend to infinity in opposite directions. Horizontal asymptotes always follow the formula y = C, while vertical asymptotes will always follow the similar formula x = C, where the value C represents any constant. Finding asymptotes, whether those asymptotes are horizontal or vertical, is an easy task if you follow a few steps.
Vertical Asymptotes: First Steps
To find a vertical asymptote, first write the function you wish to determine the asymptote of. Most likely, this function will be a rational function, where the variable x is included somewhere in the denominator. As a rule, when the denominator of a rational function approaches zero, it has a vertical asymptote. Once you've written out your function, find the value of x that makes the denominator equal to zero. As an example, if the function you're working with is y = 1/(x+2), you would solve the equation x+2 = 0, an equation which has the answer x = -2. There may be more than one possible solution for more complex functions.
Finding Vertical Asymptotes
Once you've found the x value of your function, 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. If you're working with a more complex function that has more than one possible solution, you'll need to take the limit of each possible solution. Finally, write the equations of the function's vertical asymptotes by setting x equal to each of the values used in the limits. For this example, there is only one asymptote: given by the equation the vertical asymptote is equal to x = -2.
Horizontal Asymptotes: First Steps
While horizontal asymptote rules may be slightly different than those of vertical asymptotes, the process of finding horizontal asymptotes is just as simple as finding vertical ones. Begin by writing out your function. Horizontal asymptotes can be found in a wide variety of functions, but they will again most likely be found in rational 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 (because 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) is equal to 1.
Finding Horizontal Asymptotes
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.
Finding Asymptotes for Trigonometric Functions
When dealing with problems with trigonometric functions that have asymptotes, don't worry: finding asymptotes for these functions is as simple as following the same steps you use for finding the horizontal and vertical asymptotes of rational functions, using the various limits. However, when attempting this it is important to realize that trig functions are cyclical, and as a result may have many asymptotes.
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