plane = ParametricPlot3D[{Cos[t],3Exp[2t],3Exp[-2t]},{t,-1,1},PlotStyle->{Red,Thick}]

tangentlines= {Cos[t]-Sin[t] s,3Exp[2t] + 6 Exp[2t] s,3Exp[-2t]-6 Exp[-2t] s}

tan1 = tangentlines /. t-> -1/2
tan2 = tangentlines /. t-> 0
tan3 = tangentlines /. t-> 1/2

listtangentlines = {tan1,tan2,tan3}

tangents = ParametricPlot3D[Evaluate[listtangentlines],{s,-1,1},PlotStyle->{{Blue,Thick}}]

Show[plane,tangents]