(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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This limit will not exist for every ", StyleBox["x", "Input"], ". For example, ", StyleBox["f[1] = 1", "Input"], ", but ", StyleBox["f[n,2]", "Input"], " would quickly go to infinity as ", StyleBox["n", "Input"], " increases. 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The key to understanding ", StyleBox["f[x]", "Input"], " is to realize that if this limit exists then the recursive definttion\n", StyleBox[" f[n,x] = x^f[n-1,x]\n", "Input"], "implies\n", StyleBox[" f[x] = x^f[x]\n", "Input"], "Let ", StyleBox["y = f[x]", "Input"], ". Then ", StyleBox["y = x^y", "Input"], " and we can solve: ", StyleBox["x = y^(1/y)", "Input"], ". The function ", StyleBox["y^(1/y)", "Input"], " is the inverse of ", StyleBox["f[x]", "Input"], ", and vice-versa, at least on intervals containing ", StyleBox["1", "Input"], " where these functions are 1-1. 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