Magnetic field at the center of circular current carrying loop is given by
$$ B=\frac{\mu I}{2 R} $$
Where $\mu$ is the permeability of free space and $R$ is the radius of loop.
In a question by calculation I got a field at the center of the regular $n$-sided polygon as
$$ B=\frac{\mu I n \sin \frac \pi n} {2 \pi R} $$
Now the question also asks about what when $n$ goes to infinity the polygon is a circle.
With some assumptions
- $n$ goes to infinity, hence $\theta$ goes to zero
- hence $\sin \theta=\theta$
then we get the magnetic field same as a circle.
My question is when we derive $B$ directly assuming a circle we get an answer. when we derive $B$ of polygon and assume $n=\infty$ and do approximation, why do we get the same correct answer since there is an approximation involved.