Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family of degree $2$ maps defined (for $t$ small and non zero) by $$u_t([X,Y]) := [X^2, t Y^2, XY].$$ Note that as $t$ goes to zero, $u_t$ converges outside $p:= [0,1]$ to the degree one map $v_1$ given by $$ v_1([X,Y]):= [X,0,Y].$$ Note that the map $u_t$ is an immersion for all $t \neq 0$.
$\textbf{Question:}$ Suppose $p_{s} :=[s,1]$ is a point close to $p = [0,1]$ (i.e. $s$ is small). Is it correct to say that for small $t$ and $s$, the map $u_t$ $\textbf{very nearly}$ fails to be an immersion at $p_s$? The word ''very nearly'' is to be made sense in some appropriate metric.
My reason for asking this question is as follows. Suppose we denote the coordinates in $\mathbb{P}^2$ by $[x,y,z]$. The sequence of maps $u_t$ has the image defined by the curve $$ xy -t z=0.$$ Naively we expect a pair of straight lines in the limit as $t$ goes to zero. This idea can be made very precise using the idea of Gromov Convergence (as explained in the book "J-Holomorphic Curves and Symplectic Topology" by McDuff and Salamon).
The precise statement is that the limit of the curves $u_t$ converges to a stable map whose domain is a wedge of two spheres (i.e. a nodal riemann surface). On the first component define the map to be $v_1([X,Y]):= [X,0,Y]$. On the second sphere define the map to be $v_2([X,Y]):= [0,Y,X]$. And identify the first and the second sphere at $[0,1]$ and $[1,0]$ respectively.
Hence it seems like the sequence $u_t$ should in some sense be giving a smoothing of the final domain (a wedge of two spheres) and that sequence should be very close to not being an immersion near the "bad" point $[0,1]$ (or more precisely the bubbling point).
But I am unable to see this by actually writing down an explicit formula.
$\textbf{Remark added later:}$ I believe I was mistaken in asking this question. The bubbling points are infact the places where the derivative actually blows up (certainly doesn't go to zero). The correct example to consider is $$w_{t}[X,Y] := [t X^2, tY^2, XY] $$
The bubbling points are $[1,0]$ and $[0,1]$. It is very easy to see that the derivative of $w_t$ is very close to zero around any point that is away from $[1,0]$ or $[0,1]$ (if $t$ is close to zero). I think that makes sense geometrically, because the whole energy of $w_t$ gets concentrated around the two bubbling points. In a neighbourhood of any non bubbling point, the map $w_t$ gradually looks like a constant map and hence very nearly fails to be an immersion.
