(*f[x_]:=5x-3;x0=1;L=2;delta=epsilon/5;*)

(***** define your function, x0, L, epsilon here ***)
f[x_]:=x^2;x0=2;L=4;
epsilon=0.01;delta=Min[x0-Sqrt[L-epsilon],Sqrt[L+epsilon]-x0];
(***** end of define your function, x0, L, epsilon here ***)

x=RandomReal[{x0-delta,x0+delta}];
Print["{epsilon,de lta,x,f[x]-L} =",{delta,x,f[x]-L,epsilon}];
Print["Is {f[x]-L|<espilon? ",Abs[f[x]-L]<epsilon];
pltf=Plot[f[x],{x,x0-delta,x0+delta},PlotRange->{f[x0-5delta],f[x0+5delta]},AxesLabel->{"x","y"},
PlotLabel->{"x0=",x0," delta=",FortranForm[delta]," x=",FortranForm[x]}
];
pltLp=Plot[L+epsilon,{x,x0-5delta,x0+5delta},Filling->L-epsilon];
pltLm=Plot[L-epsilon,{x,x0-5delta,x0+5delta}];
gl=Graphics[Line[{{x0,f[x0-5delta]},{x0,f[x0+5delta]}}]];
glx=Graphics[{Dashed,Line[{{x,0.5f[x]},{x,f[x]},{x,1.5f[x]}}]}];
gp=ListPlot[{{x,f[x]}},PlotMarkers->Style["×",Large,Blue]];
gpl=ListPlot[{{x0+delta,f[x0+delta]},{x0-delta,f[x0-delta]}},PlotMarkers->Style["⊗",Large,Red]];
show=Show[pltf,pltLp,pltLm,gl,gp,gpl,glx];
Print[show];
Print[" "];