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

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:

\[\frac{\tan x}{x} =\frac{\sin x}{x \cdot \cos x}=\frac{\sin x}{x} \times \frac{1}{\cos x} \\\] \[\begin{aligned} \lim_{x\to 0} \frac{\tan x}{x} &= \lim_{x\to 0} \left(\frac{\sin x}{x} \times \frac{1}{\cos x}\right)\\ &=1 \times 1\\ &=1 \end{aligned}\]we conclude that:

\[\lim _{x \rightarrow 0} \frac{\tan x}{x} = 1\]If you found this post or this website helpful and would like to support our work, please consider making a donation. Thank you!

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