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L-lim-x-0-x-1-x-




Question Number 137523 by SOMEDAVONG last updated on 03/Apr/21
L=lim_(x→0) ((x!−1)/x)
$$\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}!−\mathrm{1}}{\mathrm{x}} \\ $$
Answered by Dwaipayan Shikari last updated on 03/Apr/21
lim_(x→0) ((Γ(x+1)−1)/x)=((1−γx−1)/x)=−γ  Γ(x+1)∼1−γx
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\Gamma\left({x}+\mathrm{1}\right)−\mathrm{1}}{{x}}=\frac{\mathrm{1}−\gamma{x}−\mathrm{1}}{{x}}=−\gamma \\ $$$$\Gamma\left({x}+\mathrm{1}\right)\sim\mathrm{1}−\gamma{x} \\ $$

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