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Asymptotes

An asymptote to a curve is a straight line which the curve approaches without crossing it. If we go sufficiently far along the line, the curve becomes arbitrarily close.

A simple example is the graph of y = 1/x. This curve has both the x-axis and the y-axis as asymptotes.

As x goes to infinity, the graph approaches the x-axis from above; as x goes to negative infinity, the graph approaches the x-axis from below.

Similarly, as y goes to infinity, the graph approaches the y-axis from the right; as y goes to negative infinity, the graph approaches the y-axis from the left.

Diagonal asymptotes are also possible; for instance, the graph of y = (1/x) + x has the line y = x as an asymptote. (The y-axis is also an asymptote.)