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Factoring stuff out of limits

daigo

Member
Jun 27, 2012
60
If I do this:

[TEX]\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}[/TEX]

Am I allowed to do this?

[TEX]\frac{5}{2} \cdot \lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\alpha} = \frac{5}{2} \cdot 1 = \frac{5}{2}[/TEX]

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

Member
Jun 27, 2012
60
Also I didn't want to create another thread since this is kind of relevant:

If I have:

[tex]\lim_{\alpha \rightarrow 0} \frac{\sin ^{2}\alpha }{\alpha^{2}} = \lim_{\alpha \rightarrow 0} (\frac{\sin\alpha }{\alpha})^{2}[/tex]

Can I do this?

[tex]\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[/tex]
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
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).