Covariance of a random variable and a converging random variable.

Question

If I have $\underset{n\rightarrow\infty}{\lim}X_n=X$. ($X_n$ converges in distribution to $X$). Does $\underset{n\rightarrow\infty}{\lim}$Cov$(X_n,X)$=Var$(X)$?

To me it should, if I am allowed to pass the limit into the covariance.

Answer 1

Even if the distribution of $X_n$ is exactly the distribution of $X$, this need not be true. Consider $X\sim N(0,1)$ and $X_n =-X$.