"how to solve this limit limθ→0tan⁡(5θ)tan⁡(10θ)" was answered by our experts

Jaylin Clements

Answered question

2022-03-29

how to solve this limit

lim_{θ \to 0} \frac{\tan (5 θ)}{\tan (10 θ)}

glikozyd3s68

Beginner2022-03-30Added 16 answers

lim_{θ \to 0} \frac{\tan 5 θ}{\tan 10 θ} = lim_{θ \to 0} (\frac{\sin 5 θ}{5 θ} \cdot \frac{10 θ}{\sin 10 θ} \cdot \frac{5 θ}{10 θ} \cdot \frac{\cos 10 θ}{\cos 5 θ}) = \frac{1}{2}

zevillageobau

Beginner2022-03-31Added 13 answers

Call

x = 5 θ

and note that

lim_{θ \to 0} \frac{\tan 5 θ}{\tan 10 θ} = lim_{x \to 0} \frac{\tan x}{\tan 2 x} = lim_{x \to 0} \frac{\tan x}{2 \frac{\tan x}{1 - \tan^{2} x}}

= \frac{1}{2} lim_{x \to 0} (1 - \tan^{2} x) = \dots

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