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Question Number 142309 by qaz last updated on 29/May/21

$$\underset{\mathrm{x}\to 0^{+}}{\lim }\frac{{\mathrm{x}}^{{\left(\sin \mathrm{x}\right)}^{\mathrm{x}}}-{\left(\sin \mathrm{x}\right)}^{{\mathrm{x}}^{\sin \mathrm{x}}}}{{\mathrm{x}}^3}=?$$

Answered by mnjuly1970 last updated on 29/May/21

$${lim}_{x\to 0}{\left(sin\left(x\right)\right)}^x={lim}_{x\to 0}{\left(x\right)}^{sin\left(x\right)}=1$$$$A:={lim}_{x\to 0}{\left(sin\left(x\right)\right)}^x$$$$log\left(A\right)={lim}_{x\to 0}xlog\left(sin\left(x\right)\right)\stackrel{hop}{=}0$$$$A=e^0=1$$$$similarly{lim}_{x\to 0}{\left(x\right)}^{sin\left(x\right)}=1$$$${lim}_{x\to 0}\frac{x-sin\left(x\right)}{x^3}\underset{expansion}{\stackrel{maclaurin}{=}}{lim}_{x\to 0}\frac{x-(x-\frac{x^3}{3!})}{x^3}$$$$\mathit{\varphi }:=\frac{1}{6}....✓$$

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