An asymptote is a line that a curve approaches but never actually touches. It can be horizontal, vertical, or oblique, depending on the behavior of the function as it extends towards infinity. Asymptotes help in understanding the end behavior of graphs, particularly in rational functions where the numerator and denominator can lead to undefined values. In mathematics, asymptotes are important for sketching graphs and analyzing functions. For example, in the case of the function f(x) = 1/x, the vertical asymptote occurs at x = 0, while the horizontal asymptote is at y = 0. These lines indicate where the function's values become very large or very small.