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Question Number 130854 by mathlove last updated on 29/Jan/21

$$\underset{x\to 2}{\lim }\frac{x^x\cdot x!-4(3x-4)!}{3x!-6}$$

Answered by Dwaipayan Shikari last updated on 29/Jan/21

$$\underset{x\to 2}{\lim }\frac{x^x(1+logx)x!+x^{x}x!\psi (x+1)-12(3x-4)!\psi (3x-3)}{3x!\psi (x+1)}$$$$=\frac{4(1+log\left(2\right))+4(\frac{3}{2}-\gamma )-24(\frac{3}{2}-\gamma )}{9-6\gamma }$$$$=\frac{4log\left(2\right)-26+20\gamma }{9-6\gamma }$$

Commented by mathlove last updated on 29/Jan/21

$$\psi and\gamma isanyfunction$$

Commented by Dwaipayan Shikari last updated on 29/Jan/21

$$\frac{{\mathrm{\Gamma }}^{\prime }\left(x\right)}{\mathrm{\Gamma }\left(x\right)}=\psi \left(x\right)\left(Digammafunction\right)$$$$\psi \left(x\right)=-\gamma +\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{n+1}-\frac{1}{n+x}$$$$\mathrm{\Gamma }\left(x\right)={\int }_0^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}dt=(x-1)!$$$$\gamma =\mathcal{E}ulerianConstant=\underset{n\to \mathrm{\infty }}{\lim }\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{1}{k}-log\left(n\right)$$

Commented by mathlove last updated on 29/Jan/21

$$thanksbro$$

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