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Question Number 82920 by abdomathmax last updated on 25/Feb/20
$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{ln}\left(\mathrm{1}+{ln}\left(\mathrm{1}+{x}\right)\right)}{{x}} \\ $$
Answered by mind is power last updated on 25/Feb/20
$${ln}\left(\mathrm{1}+{x}\right)\sim{x} \\ $$$$\Rightarrow{ln}\left(\mathrm{1}+{ln}\left(\mathrm{1}+{x}\right)\right)\sim{ln}\left(\mathrm{1}+{x}\right) \\ $$$$\frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}}\rightarrow\mathrm{1} \\ $$$$\Rightarrow\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{1}+{ln}\left(\mathrm{1}+{x}\right)\right)}{{x}}\rightarrow\mathrm{1} \\ $$