Is factoring out of limits allowed?

[hatelove]

If I do this:

$$\lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\frac{2\alpha}{5}} = \lim_{\alpha\rightarrow 0} \frac{5sin\alpha}{2\alpha} = \lim_{\alpha\rightarrow 0} \frac{5}{2}\cdot \frac{sin\alpha}{\alpha}$$

Am I allowed to do this?

$$\frac{5}{2} \cdot \lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\alpha} = \frac{5}{2} \cdot 1 = \frac{5}{2}$$

Or did I do something in the very first steps incorrectly?

 

[hatelove]

Also I didn't want to create another thread since this is kind of relevant:

If I have:

$$\lim_{\alpha \rightarrow 0} \frac{\sin ^{2}\alpha }{\alpha^{2}} = \lim_{\alpha \rightarrow 0} (\frac{\sin\alpha }{\alpha})^{2}$$

Can I do this?

$$\lim_{\alpha \rightarrow 0} (\frac{\sin\alpha }{\alpha} \cdot \frac{\sin\alpha }{\alpha}) = (\lim_{\alpha \rightarrow 0} \frac{\sin\alpha }{\alpha}) \cdot (\lim_{\alpha \rightarrow 0} \frac{\sin\alpha }{\alpha}) = 1 \cdot 1 = 1$$

 

[Ackbach]

You can do all of those things... provided that your final limits exist. They do in this case, so I would say you're on safe ground. There are limit theorems that allow you to distribute the taking of a limit with addition, subtraction, multiplication, division (no divide by zero, of course).