Factorial Limit

Calculus Level 2

Let f ( x ) = ( x ! ) 2 ( 2 x ) ! f(x) = \dfrac{(x!)^2}{(2x)!} , find lim ⁡ x → ∞ f ( x ) \displaystyle \lim_{x\to\infty} f(x) .

0 1 ∞ 0.4 0.2

Ar Hakim by AR Hakim , 25, Indonesia

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1 solution

lim ⁡ x → ∞ ( x ! ) 2 2 x ! = lim ⁡ x → ∞ 1 ( 2 x x ) = 0 \lim_{x\to \infty} \frac{(x!)^2}{2x!} = \lim_{x\to \infty} \frac{1}{\binom{2x}{x}} = 0

due to ( 2 x x ) = number of sets of cardinal x what may be formed from a set of cardinal 2x \binom{2x}{x} = \text{number of sets of cardinal x what may be formed from a set of cardinal 2x}

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