A function or sequence diverges if its values do not approach a fixed value.

This can happen in one of two ways:

• The values can head off to infinity or minus infinity (get larger and positive or larger and negative)
• The values can oscillate up and down without ever settling on a value that they will approach.

For example, \(y = x^2\) diverges. Its values approach infinity as \(x\) gets very large and positive.

For example, \(\sin (x)\) also diverges. Its values wobble around between \(-1\) and \(1\), so there is no fixed value that they approach as \(x\) approaches infinity.

Finally, \(x \sin (x)\) also diverges. The oscillations get larger and larger as \(x\) gets larger.