I know that I can substitute $2x$ for $x$ in 

$$\displaystyle\lim\_{x \to 0} \frac{\sin x}{x} = 1$$

to obtain the following equality:

$$\displaystyle\lim\_{2x \to 0} \frac{\sin 2x}{2x} = 1$$

But the slide below seems to suggest that 

$$\displaystyle\lim\_{2x \to 0} \frac{\sin 2x}{2x} = 1 \Longleftrightarrow \displaystyle\lim\_{x \to 0} \frac{\sin 2x}{2x} = 1 \tag{1}$$

(note that $2x \to 0$ becomes $x \to 0$). 

What's the justification for this?

Furthermore, what's the justification for 

$$ \lim\_{x \to 0} \left( 2 \frac{\sin(2x)}{2x}\right) = 2 \cdot \lim\_{x \to 0} \frac{\sin(2x)}{2x} \tag{2}$$ ?

[![enter image description here](https://i.stack.imgur.com/63iEF.png)](https://i.stack.imgur.com/63iEF.png)