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Question-202530




Question Number 202530 by Calculusboy last updated on 28/Dec/23
Answered by MathematicalUser2357 last updated on 29/Dec/23
=lim_(x→0) ((log(1+x)^(1+x) −x)/x^2 )  =(1/2)lim_(x→0) ((∂^2 /∂^2 x)(log(1+x)^(1+x) −x))  = ...(15∼30 steps)...  =⟨LIMIT DOES NOT CONVERGE⟩
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{log}\left(\mathrm{1}+{x}\right)^{\mathrm{1}+{x}} −{x}}{{x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\partial^{\mathrm{2}} }{\partial^{\mathrm{2}} {x}}\left(\mathrm{log}\left(\mathrm{1}+{x}\right)^{\mathrm{1}+{x}} −{x}\right)\right) \\ $$$$=\:…\left(\mathrm{15}\sim\mathrm{30}\:\mathrm{steps}\right)… \\ $$$$=\langle\mathrm{LIMIT}\:\mathrm{DOES}\:\mathrm{NOT}\:\mathrm{CONVERGE}\rangle \\ $$

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