How many sides does a circle has.

For those who are thinking that the answer is lim𝑛→∞𝑛=∞
lim
n
→
∞
n
=
∞
, via:

An 𝑛
n
-gon has 𝑛
n
sides;
A circle is a limit of a 𝑛
n
-gon as 𝑛→∞
n
→
∞
;
Therefore a circle has lim𝑛→∞𝑛=∞
lim
n
→
∞
n
=
∞
sides;
I'd like to mention: it's not so straightforward. If taking limits in this way were legitimate then we can show that e.g. a square has an infinite number of sides.

Consider a staircase with 𝑛
n
steps, and each step has height 1/𝑛
1
/
n
and width 1/𝑛
1
/
n
. It consists of 2𝑛
2
n
line segments. As 𝑛→∞
n
→
∞
, the staircase converges to a single line segment (i.e. the limit agrees point-for-point with a single line segment).

If we glue four of these staircases together, and take their limit, we obtain a square, which would have lim𝑛→∞4×2𝑛=∞
lim
n
→
∞
4
×
2
n
=
∞
sides.