= Solution
Let
$$
g(a,c,b)=\frac{c}{\sqrt{ab}}=\rho.
$$
Its gradient, in the coordinate order $(a,c,b)$, is
$$
\nabla g=
\left(-\frac{\rho}{2a},\frac1{\sqrt{ab}},-\frac{\rho}{2b}\right)^T.
$$
Apply the <Delta method> to the limit in part (c). Multiplying the displayed covariance matrix there on both sides by this gradient gives
$$
(\nabla g)^TV\nabla g=(1-\rho^2)^2.
$$
Consequently the <asymptotic distribution of the Gaussian sample correlation> is
$$
\boxed{
\sqrt n(\widehat\rho-\rho_{X,Y})
\xrightarrow d
N\!\left(0,
\left(1-\frac{\Sigma_{12}^2}{\Sigma_{11}\Sigma_{22}}\right)^2
\right).}
$$
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