Suppose we have two correlated Normal variables $X_A$ and $X_B$, with respective standard deviation $A$ and $B$, and correlation $\rho$. The variable $X_A + X_B$ has a standard deviation in excess (or deficiency) of that of $X_A$ given by: $$ E = \sqrt{A^2 + 2 \rho A B + B^2} - A $$ My question is: as $A$ becomes large (with $A >> B$), show rigorously that $E \rightarrow \rho B$.

This is straightforward to verify for $\rho = \pm 1$, but the intermediate case eludes me.