Question Number 167662 by mathlove last updated on 22/Mar/22
![lim_(x→3) [((log((2log(√(x^3 +2))))^(1/3) ))^(1/4) ]=?](Q167662.png)
$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\left[\sqrt[{\mathrm{4}}]{{log}\sqrt[{\mathrm{3}}]{\mathrm{2}{log}\sqrt{{x}^{\mathrm{3}} +\mathrm{2}}}}\right]=? \\ $$
Answered by alephzero last updated on 23/Mar/22
$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\sqrt[{\mathrm{4}}]{\mathrm{ln}\:\sqrt[{\mathrm{3}}]{\mathrm{2ln}\:\sqrt{{x}^{\mathrm{3}} +\mathrm{2}}}}\:= \\ $$$$=\:\mathrm{lim}\sqrt[{\mathrm{4}}]{\frac{\mathrm{1}}{\mathrm{3}}\mathrm{ln}\:\mathrm{ln}\left({x}^{\mathrm{3}} +\mathrm{2}\right)}\:= \\ $$$$=\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{1}}{\mathrm{3}}\mathrm{ln}\:\mathrm{ln}\:\mathrm{29}} \\ $$