The Function

n[x_] = Sin[Tan[x]] - Tan[Sin[x]];
d[x_] = 
        ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]];
f[x_] = n[x]/d[x];

Part 1

\(N[Table[{n, \*""f[\!\(10\^\(-n\)\)]"", f[10\^\(-n\)]}, {n, 0, 
            4}], 25] // TableForm\)

Part 2

Off[Power::"infy"]; Off[Plot::"plnr"]; 
    Off[∞::"indet"];
Plot[f[x], {x, -0.99, 0.99}]; 
    Plot[f[x], {x, -0.0001, 0.0001}, 
      PlotPoints -> 1000];

Part 3: l'Hospital's Rule

Table[{m, 
          FullSimplify[D[n[x], {x, m}]]/FullSimplify[D[d[x], {x, m}]]}, {m, 
          0, 2}] // TableForm

Part 4

Series[n[x], {x, 0, 25}]

Series[d[x], {x, 0, 25}]

Part 5

Limit[f[x], x -> 0]