Was reading Thomas Calculus and came across the limit of $\frac{\sin(x)}{x}$ at $x \rightarrow 0$.
The method they used was comparing areas of sector and $2$ triangles to prove the inequality
$\sin(x) < x < \tan(x)$,
and then I remembered one of the proofs of area of circles which was done using joining $n$ triangles and so basically they used $\frac{\sin(x)}{x} \rightarrow 1$ as $x \rightarrow 0$.
Isn't it wrong to use area of circle to prove the limit?