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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:

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