F[x_]:=(x^2-1)/(x^3-1)

For[n=1,n<6,n++,Print[N[1+10^-n],"  ",N[F[1+10^(-n)]]]]

Plot[F[x],{x,0,2}]

H[x_]:=x+x/Abs[x]

For[n=1,n<6,n++,Print[N[10^-n],"  ",N[H[10^(-n)]]]]

For[n=1,n<6,n++,Print[N[-10^-n],"  ",N[H[-10^(-n)]]]]

Plot[H[x],{x,-1,1}]

G[x_]:=1/x

For[n=1,n<6,n++,Print[N[10^-n],"  ",N[G[10^(-n)]]]]
For[n=1,n<6,n++,Print[N[-10^-n],"  ",N[G[-10^(-n)]]]]

Plot[G[x],{x,-1,1}]

S[x_]:=Sin[2*Pi/x]

For[n=1,n<6,n++,Print[N[10^-n],"  ",N[S[10^(-n)]]]]
For[n=1,n<6,n++,Print[N[-10^-n],"  ",N[S[-10^(-n)]]]]

For[n=1,n<6,n++,Print[N[(2/(1/2 +10^n))],"  ",N[S[2/(1/2 +10^n)]]]]

Plot[S[x],{x,-1,1}]