What is the definition of an asymptote?

An asymptote is defined as a line that a graph approaches but never actually reaches. This concept is significant in the study of graphs, particularly in relation to rational functions, exponential functions, and logarithmic functions. The existence of asymptotes provides important information about the behavior of a graph as it moves towards certain values, typically at infinity or where the function is undefined.

For instance, vertical asymptotes can indicate where a function increases or decreases without bound as it approaches a certain x-value, while horizontal asymptotes reveal the behavior of a function as the input values become very large or very small. Understanding asymptotes is vital for grasping the overall shape and trends of graphs, as they help to illustrate limiting behavior and extreme cases of a function's outputs.