A limit for two correlated variables [on hold]

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.

put on hold as off-topic by Ben McKay, Neil Hoffman, Pace Nielsen, Boris Bukh, Mark Wildon2 days ago

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• You may multiply and divide this expression by the conjugate $\sqrt{A^2 +2\rho AB+B^2 }+A$ (which itself behaves like $2A$). – Fedor Petrov Jan 5 at 8:34