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When plotting the graph of $\sin(x^x)$ I noticed that there is no plot from about $x=143$. I don't suppose there is a purely mathematical explanation for this?

So, why is there no graph in Mathematica from $x=143$? Some kind of overflow?

f[x_] = Sin[x^x]
Plot[{f[x]}, {x, 142, 144}, PlotStyle -> Thin, PlotLegends -> "Expressions"]

Mathematica graphics

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Normally Plot uses machine precision numbers; your $x^x$ expression is hitting the limit of the numbers that can be represented in machine precision right about $x>143$.

Note:

Solve[$MaxMachineNumber == x^x, x]

(* Out: {{x -> 143.016}} *) 

You can increase the WorkingPrecision setting for Plot adequately, and the plot will be complete:

f[x_] = Sin[x^x]
Plot[{f[x]}, {x, 142, 144}, PlotStyle -> Thin, PlotLegends -> "Expressions", 
      WorkingPrecision -> 450]

http://i.stack.imgur.com/mSkAJ.png

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