Specific integral with continuous function Let $f: [0,1] \times [0,1] \to \mathbb{R}$ be a continuous function. How can I compute:

$$ \lim_{n \to \infty} \left( \frac{(2n+1)!}{(n!)^2} \right)^2 \int_{0}^{1} \int_{0}^{1} (xy(1-x)(1-y))^n f(x,y)\, dx\, dy$$
?
$$\iint_{[0,1]^2} x^n (1-x)^n y^n (1-y)^n \,dx\,dy = \left(\int_{0}^{1}z^{n}(1-z)^n\,dz\right)^2=\left(\frac{n!^2}{(2n+1)!}\right)^2$$
hence the limit, by convergence in distribution, is just $\color{red}{f\left(\frac{1}{2},\frac{1}{2}\right)}.$
HINT: Expand $f(x,y)$ in a power series $f(x,y)= \sum_{i,j=0}^\infty a_{ij}x^iy^j$. Then use the Definition of the Beta function.

For computing the Limit $n \rightarrow \infty$ you must use $\Gamma(n)=(n-1)!$ for natural number $n$.