we just need this little fact:
[math]\underset{x\to 0}{\mathop{\lim }}\,x\ln x=0,[/math] thus
[math]x^{\sin x}=\exp (\sin (x)\ln x)=\exp \left( \frac{\sin x}{x}\cdot x\ln x \right)\to 1[/math] as [math]x\to0.[/math]

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