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Let $a$ and $b$ be two positive real numbers satisfying $(a-b)^2=4(ab)^3$. Find the minimal value of $\frac{1}{a}+\frac{a}{b}$.

Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.

Show that $$\lim_{n\to\infty}\int_0^1 x^n(1-x)^n dx = 0$$