Limit approaches 0 for (tan theta/3)/theta (1 Viewer)

Najji · Jul 14, 2013

Can't seem to solve this question :/. Any help will be greatly appreciated.

HeroicPandas · Jul 14, 2013

Najji said:
Can't seem to solve this question :/. Any help will be greatly appreciated.

$\lim_{x\rightarrow 0} \frac{tan\left ( \frac{\theta}{3} \right )}{\theta} \\ \\ $Apply the rule $\lim_{\alpha\rightarrow 0} \frac{tan\alpha}{\alpha} = 1 \\ $This can proven by doing this: \\ \\ $\lim_{\alpha\rightarrow 0} \frac{\frac{sin\alpha}{cos\alpha}}{\alpha} = \lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\alpha cos\alpha} = \lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\alpha}\cdot \frac{1}{cos\alpha} = 1\cdot 1 = 1 \left ( $As $\lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\alpha} = 1\right )) \\ \\ \lim_{x\rightarrow 0} \frac{tan(\frac{\theta}{3})}{\theta} = \lim_{x\rightarrow 0} \frac{tan(\frac{\theta}{3})}{\frac{\theta}{3}} \cdot \frac{1}{3} ($Because u want $\frac{tan(same)}{same}, $then change this new form back into the question so that u dont change the question) \\ \\ \therefore = 1\cdot \frac{1}{3} = \frac{1}{3}$

Najji · Jul 14, 2013

I was never aware of the rule tan(same)/same = 1. Thanks for clarifying it Panda!

TheMathStudent · Jul 15, 2013

That's mostly because you can use tan as follows :

$\lim_{\theta \to 0} \frac{\tan \theta}{\theta} =\lim_{\theta \to 0} \frac{\sin \theta}{\theta \cos{\theta}}=\lim_{\theta \to 0} \frac{\sin \theta}{\theta}\times \frac{1}{\cos \theta}= 1 \times 1 =1$

Limit approaches 0 for (tan theta/3)/theta (1 Viewer)

Najji

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HeroicPandas

Heroic!

Najji

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TheMathStudent

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