(*^

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:[font = special1; inactive; preserveAspect; ]
                          Estimation and Limits -- Composing Functions

The following cell illustrates how Mathematica  can be used to compare two different functions.  
;[s]
4:0,1;70,0;107,2;118,0;170,-1;
3:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; preserveAspect; ]
f[x_] := x^2
g[x_] := x - 1

Plot[{f[g[x]], g[f[x]]}, {x, -3, 3},
     PlotStyle -> {{RGBColor[1, 0, 0]},
                   {RGBColor[0, 0, 1]}}] 
:[font = special1; inactive; preserveAspect; ]
The following cell illustrates how Mathematica  can be used to determine a limit experimentally.  In this example we are trying to determine the limit as   x --> 4   of   (x^2 - 16) / (x - 4)
;[s]
6:0,0;35,1;46,0;156,2;163,0;170,2;192,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;2,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect; ]
f[x_] := (x^2 - 16)/(x - 4)

f[4.001]
f[4.00001]

f[3.99]
f[3.9999]
:[font = special1; inactive; preserveAspect; ]
The following cell illustrates the way that Mathematica   can be used to solve equations.  This is useful for the  Check Your Understanding  problem.  Evaluate the next cell now. 
;[s]
7:0,0;44,1;55,0;115,2;139,0;151,3;178,0;180,-1;
4:4,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
G[m_] := m (1 - 1/(2^m))
FindRoot[G[m] == 1, {m, 1}]
^*)