Binomials and Limits

Calculus Level 3

Define a relation \(f:\Bbb{Z^+}\mapsto\Bbb{R^+}\) given by:

\[f(n)=4^n\cdot \frac{\dbinom{2n}{n}}{\dbinom{4n}{2n}}\]

Find the value of the following limit:

\[\lim_{n\to\infty}f(n)\]

This problem is not original. It is taken from a question posted on MSE.
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