# how to solve this limit $$\displaystyle\lim_{{\theta\to{0}}}{\frac{{{\tan{{\left({5}\theta\right)}}}}}{{{\tan{{\left({10}\theta\right)}}}}}}$$

how to solve this limit $\underset{\theta \to 0}{lim}\frac{\mathrm{tan}\left(5\theta \right)}{\mathrm{tan}\left(10\theta \right)}$
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glikozyd3s68
$\underset{\theta \to 0}{lim}\frac{\mathrm{tan}5\theta }{\mathrm{tan}10\theta }=\underset{\theta \to 0}{lim}\left(\frac{\mathrm{sin}5\theta }{5\theta }\cdot \frac{10\theta }{\mathrm{sin}10\theta }\cdot \frac{5\theta }{10\theta }\cdot \frac{\mathrm{cos}10\theta }{\mathrm{cos}5\theta }\right)=\frac{1}{2}$
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zevillageobau
Call $x=5\theta$ and note that
$\underset{\theta \to 0}{lim}\frac{\mathrm{tan}5\theta }{\mathrm{tan}10\theta }=\underset{x\to 0}{lim}\frac{\mathrm{tan}x}{\mathrm{tan}2x}=\underset{x\to 0}{lim}\frac{\mathrm{tan}x}{2\frac{\mathrm{tan}x}{1-{\mathrm{tan}}^{2}x}}$
$=\frac{1}{2}\underset{x\to 0}{lim}\left(1-{\mathrm{tan}}^{2}x\right)=\dots$