Define asymotote and explain the relationship between a hyperbola and its asymptotes?

Answer 1

An asymptote of a curve is a line where the distance of the curve and line approach zero as they tend to infinity (they get closer and closer without ever meeting)

If one zooms out of a hyperbola, the straight lines are usually asymptotes as they get closer and closer to a specific point, yet do not reach that point.

Answer 2

The asymptotes of a curve are straight lines such that, as one of the curve's coordinates becomes infinitely large, the curve comes infinitesimally close to the line without ever reaching it.
A hyperbola has two asymptotes which intersect at its centre of symmetry. These divide the coordinate plane into four segments and the two arms of the hyperbola are contained within diagonally opposite segments.