What is an asymptote?

An asymptote is indeed characterized as a line that a graph approaches as the values of the function extend towards positive or negative infinity, but never actually intersects or touches. This concept is crucial in understanding the behavior of functions, particularly in calculus and graphing.

In many cases, asymptotes can be either vertical, horizontal, or oblique. A vertical asymptote indicates that as you approach a certain x-value, the function's values head towards infinity or negative infinity. Horizontal asymptotes describe the behavior of a function as the x-values approach infinity, signaling that the function will get closer and closer to a specific y-value without actually reaching it.

The roles of asymptotes are vital in sketching graphs and analyzing the limits of functions, contributing to discussions surrounding infinity and continuity. The other options present different geometrical relationships but do not capture the essence of how asymptotes function regarding graphs.