ag.algebraic geometry – Limit of Line Bundles on Smooth Curves Degenerating to Double Line

Consider a family of smooth plane conics $f_lambda(x,y,z)=0$ as a family $T_lambda = (C,L,v_1,v_2,v_3)_lambda$ of genus zero curves with a degree 2 line bundle $L$ and an ordered basis $v_i$ for the space of global sections of $L$. Suppose that $f_lambda$ degenerates to the double line $x^2=0$ when $lambda=0.$

What are various geometric objects which could serve as a limit of $T_lambda$ as $lambda$ goes to zero, and what are some examples of situations where each limit would be (in)appropriate?

For example, the equation $x^2=0$ would be a possible limit that seems appropriate if we’re thinking about the $T_lambda$ as polynomial equations, but less appropriate if we’re thinking about the $T_lambda$ as $5$-ples $(C,L,v_1,v_2,v_3)_lambda.$