One quick question about Moore–Penrose inverse of a symmetric matrix

Question

I was wondering whether the following equation is correct, and why if so. Here the $ \left( A\right)^{+} $ is the MP inverse of matrix A.

$$ \left(A^{\top} A\right)^{+}A^{\top} =\lim _{\rho \rightarrow 0}\left(A^{\top} A+\rho I\right)^{-1} A^{\top} $$

Answer 1

No. For example, if $$ A:=\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}, $$ then the limit does not exist.