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Another type of function which is similar to composition of functions is called a continued function. Continued functions contain some sort of repeated pattern on towards infinity. One such example is

One approach to reformulating the function is to write y = 1+x/y. Afterall the pattern repeats in the denominator. Solving the resulting quadratic equation gives an expression for y = f(x) that is easy to calculate in a finite amount of time.

An important issue is whether a continued function actually converges to a finite value. For example

can be found to be

It turns out the trick to solve for the expression only works if $latex \left| x \right| < 1&s=1$

As you can see

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