Let \(N=10^{100}.\) The graph of
\[
\mathrm{f}(x)=\frac{x^{N}}{1+x^{N}}+2
\]
for \(-3\leqslant x\leqslant3\) is sketched in the following diagram.
\noindent
\psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.5pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-4.15,-1.01)(4.08,4.09) \psaxes[labelFontSize=\scriptstyle,xAxis=true,yAxis=true,labels=none,Dx=1,Dy=1,ticksize=0pt 0,subticks=2]{->}(0,0)(-4.15,-1.01)(4.08,4.09)[\(x\),140] [\(y\),-40] \psline(-4,3)(-1,3) \psline(-1,3)(-1,2) \psline(-1,2)(1,2) \psline(1,2)(1,3) \psline(1,3)(4,3) \rput[tl](-1.39,-0.2){\(-1\)} \rput[tl](1,-0.2){\(1\)} \rput[tl](0.19,1.9){\(2\)} \rput[tl](0.19,3.18){\(3\)} \end{pspicture*}
\par
Explain the main features of the sketch.
Sketch the graphs for \(-3\leqslant x\leqslant3\) of the two functions
\[
\mathrm{g}(x)=\frac{x^{N+1}}{1+x^{N}}
\]
and
\[
\mathrm{h}(x)=10^{N}\sin(10^{-N}x).
\]
In each case explain briefly the main features of your sketch.