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Question Number 129655 by liberty last updated on 17/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim }{\left(\log {}_x\left(2021x\right)\right)}^{\log x}=?$$$$\lceil \ast \rceil \left(\sqrt{{\pi }^{\ell \mathrm{iberty}}}\right)$$

Answered by bemath last updated on 17/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim }{\left(\frac{\ln \left(2021\mathrm{x}\right)}{\ln \mathrm{x}}\right)}^{\ln \mathrm{x}}=\underset{x\to \mathrm{\infty }}{\lim }{(1+\frac{\ln 2021}{\ln \mathrm{x}})}^{\ln \mathrm{x}}$$$$\underset{x\to \mathrm{\infty }}{\lim }{\left[(1+\frac{1}{\left(\frac{\ln \mathrm{x}}{\ln 2021}\right)})\frac{\ln \mathrm{x}}{\ln 2021}\right]}^{\ln 2021}={\mathrm{e}}^{\ln 2021}=2021$$

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