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Question Number 188060 by mathlove last updated on 25/Feb/23

Answered by aleks041103 last updated on 26/Feb/23

$$\underset{x\to 0}{lim}{\left(\underset{k=2}{\prod ^n}\sqrt[k]{cos\left(kx\right)}\right)}^{1/x^2}=$$$$=\underset{x\to 0}{lim}{\left(\underset{k=2}{\prod ^n}{(1-\frac{k^2{x}^2}{2}+o\left(x^2\right))}^{1/k}\right)}^{1/x^2}=$$$$=\underset{k=2}{\prod ^n}\underset{x\to 0}{lim}{(1-\frac{{kx}^2}{2}+o\left(x^2\right)+o(\frac{k^2{x}^2}{2}+o\left(x^2\right)))}^{1/x^2}=$$$$=\underset{k=2}{\prod ^n}\underset{x\to 0}{lim}{(1-\frac{{kx}^2}{2}+o\left(x^2\right))}^{1/x^2}=$$$$=\underset{k=2}{\prod ^n}{e}^{-k/2}=e^{-\frac{1}{2}\underset{k=2}{\sum ^n}k}=e^{-\frac{1}{2}(\frac{n(n+1)}{2}-1)}=$$$$=e^{-\frac{n^2+n-2}{4}}$$

Answered by aleks041103 last updated on 26/Feb/23

$$\underset{x\to 0}{lim}{\left(\underset{k=1}{\prod ^n}cos\left(kx\right)\right)}^{1/x^2}=$$$$=\underset{k=1}{\prod ^n}\underset{x\to 0}{lim}{\left(cos\left(kx\right)\right)}^{1/x^2}=$$$$=\underset{k=1}{\prod ^n}\underset{x\to 0}{lim}{(1-\frac{k^2{x}^2}{2}+o\left(x^2\right))}^{1/x^2}=$$$$=\underset{k=1}{\prod ^n}{e}^{-\frac{k^2}{2}}=exp(-\frac{1}{2}\underset{k=1}{\sum ^n}{k}^2)=$$$$=exp(-\frac{1}{2}\frac{n(n+1)(2n+1)}{6})=$$$$=e^{-\frac{n(n+1)(2n+1)}{12}}$$

Answered by aleks041103 last updated on 26/Feb/23

$$\underset{x\to 0}{lim}{\left(\frac{1}{n}\underset{k=1}{\sum ^n}{e}^{kx}\right)}^{1/x}=$$$$=\underset{x\to 0}{lim}{\left(\frac{1}{n}\underset{k=1}{\sum ^n}(1+kx+o\left(x\right))\right)}^{1/x}=$$$$=\underset{x\to 0}{lim}{\left(\frac{1}{n}(n+\left(\underset{k=1}{\sum ^n}k\right)x+o\left(x\right))\right)}^{1/x}=$$$$=\underset{x\to 0}{lim}{(1+\frac{n+1}{2}x+o\left(x\right))}^{1/x}=$$$$=e^{\frac{n+1}{2}}$$

Answered by aleks041103 last updated on 26/Feb/23

$$\underset{x\to 0}{lim}{\left(\frac{1}{n}\underset{k=1}{\sum ^n}{a}_k^x\right)}^{1/x}=$$$$=\underset{x\to 0}{lim}{\left(\frac{1}{n}\underset{k=1}{\sum ^n}{e}^{ln\left(a_k\right)x}\right)}^{1/x}=$$$$=\underset{x\to 0}{lim}{(1+ln\left(\sqrt[n]{a_1{a}_2...a_n}\right)x+o\left(x\right))}^{1/x}=$$$$=e^{ln\left(\sqrt[n]{a_1...a_n}\right)}=\sqrt[n]{\underset{k=1}{\prod ^n}{a}_k}$$

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