I would like to define an inverse on matrices whose entries may be positive or negative infinity.

To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I easily compute $\lim_{x\to\infty} (A+xB)^{-1}$?

Of course this is not so interesting if $B$ is invertible. It is easy to see that, in this case, $\frac{1}{x}A + B$ is invertible for a suitably large $x$ and the inverse converge to that of $B$. So here the limit equals to $0$.

Another special case is if $B$ is diagonal with $k>0$ zero eigenvalues, ordered to the top. Here the limit equals to a 2x2 block matrix where the first block is the inverse of $A_{11}$, i.e. the corresponding block of $A$; and the other elements are all 0.

The problem is, how do I handle the special case? When is there a limit, and what does it equal to? Has this been investigated anywhere?