What is meant by the terms vertical and horizontal asymptotes
Here is a nice graph with vertical and horizontal asymptotes. THe dotted lines are the asymptotes.
http://www.mathnstuff.com/math/spoken/here/2class/300/fx/library/234/n1oxp4.gif
Vertical and horizontal asymptotes are concepts in calculus and graph theory that describe the behavior of a function as its input approaches certain values.
A vertical asymptote is a vertical line that the graph of a function approaches but never touches. It occurs when the function approaches infinity or negative infinity as the input approaches a certain value. In the given graph, the vertical asymptotes are represented by the dashed vertical lines. As x approaches -3 and 2, the function approaches infinity or negative infinity, but it never actually reaches those values.
A horizontal asymptote, on the other hand, is a horizontal line that the graph of a function approaches as the input moves towards positive or negative infinity. In the provided graph, the horizontal asymptote is represented by the dashed horizontal line. As x approaches positive or negative infinity, the function approaches the y-value represented by the horizontal asymptote. In this case, it appears to approach the value y = 3.
To determine the presence and properties of asymptotes, you can follow these steps:
1. Examine the behavior of the function as the input approaches infinity or negative infinity. Simplify the function and determine if any terms dominate the others as x becomes very large or very small.
2. Determine the limits of the function as x approaches positive or negative infinity. This will give you the y-values for the horizontal asymptotes, if they exist.
3. Examine the behavior of the function as the input approaches certain values. Look for instances where the function approaches infinity or negative infinity without actually reaching those values. These will give you the equations for the vertical asymptotes, if they exist.
4. Plot the asymptotes on the graph.
Remember, these steps are a general guide, and each function may have its own unique characteristics.
Vertical asymptotes and horizontal asymptotes are concepts in calculus that describe the behavior of a function as the input approaches certain values or as the output approaches infinity or negative infinity.
A vertical asymptote occurs when the function approaches positive or negative infinity as the input approaches a specific value. It is a vertical line that the graph of the function gets closer and closer to, but does not intersect. In the given graph, the vertical asymptotes are indicated by the dotted lines.
On the other hand, a horizontal asymptote describes the behavior of a function as the input or output approaches positive or negative infinity. It is a horizontal line that the graph of the function approaches but does not cross. In the given graph, the horizontal asymptotes are also indicated by the dotted lines.
Asymptotes help us understand the overall behavior of a function and where it tends to go as we consider larger and larger values of the input or output.