Which of the following best captures the essence of "to infinity"?

The phrase "ad infinitum" translates from Latin to "to infinity" and is commonly used to express the idea of something continuing without end or limit. It captures the essence of an infinite process or concept, whether it be a mathematical series or a philosophical argument. This term is often seen in discussions that emphasize the boundlessness or indefinite continuation of something.

The other phrases, while also originating from Latin, have different meanings. "Ad litem" refers to being appointed for a specific legal purpose, often in a court context; "ad nauseam" indicates a discussion or argument that has been extended to the point of excess, leading to annoyance or sickening repetition; "ad hoc" denotes something that is created for a specific purpose or task without considering wider implications. Each of these phrases serves distinct functions that do not align with the concept of infinity. Therefore, "ad infinitum" stands out as the phrase that best encapsulates the meaning of "to infinity."