$_{\operatorname{Sin}2\alpha =2\operatorname{Sin}\alpha \operatorname{Cos}\alpha }^{\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}x}{x}=1}\Rightarrow \underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}2x}{4x}=\underset{x\to 0}{\mathop{\lim }}\,\frac{2\operatorname{Sin}\alpha \operatorname{Cos}\alpha }{4x}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}x}{x}\times \underset{x\to 0}{\mathop{\lim }}\,\frac{2\operatorname{Cos}x}{4}=1\times \frac{2}{4}=\frac{2}{4}=\frac{1}{2}$
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