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Proof of limit of tan x / x = 1 as x approaches 0

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How to prove that limit of tan x / x = 1 as x approaches 0 ?


Requirement

$$ \lim _{x \rightarrow 0} \frac{\sin x}{x} = 1 $$


Proof of limit of sin x / x = 1 as x approaches 0

Proof
By definition of tan x:

$$ \frac{\tan x}{x} =\frac{\sin x}{x \cdot \cos x}=\frac{\sin x}{x} \times \frac{1}{\cos x} \\ $$


(Thanks Karinou for the misprint)

$$ \begin{aligned} \lim_{x\to 0} \frac{\tan x}{x} &= \lim_{x\to 0} \left(\frac{\sin x}{x} \times \frac{1}{\cos x}\right)\\ &=1 \times 1\\ &=1 \end{aligned} $$

we conclude that:

$$ \lim _{x \rightarrow 0} \frac{\tan x}{x} = 1 $$

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  2. Proof of limit of tan x / x = 1 as x approaches 0
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