exactx[t_,ε_] :=
E^(-ε*t)Sin[Sqrt[1-ε^2]t]/Sqrt[1-ε^2];

approx1[t_,ε_] :=
(1-ε*t)Sin[t];

Clear[ε];Clear[tmax];

ε = 0.1;
tmax = 30;

exactplot =
Plot[
exactx[t,ε],{t,0,tmax},
PlotStyle->{Thick,Red},
PlotRange->All
];

approx1plot =
Plot[
approx1[t,ε],{t,0,tmax},
PlotStyle->{Thick,Blue}
];

Show[
exactplot,approx1plot,
PlotRange->All
]

approx2[t_,ε_] :=
E^(-ε*t)Sin[t];

approx2plot =
Plot[
approx2[t,ε],{t,0,tmax},
PlotStyle->{Thick,Green},
PlotRange->All
];

Show[
exactplot,approx2plot,
PlotRange->All
]