Here's the "Troll Pi" or "Pi equals 4" image.
Here's the breakdown, as simple as I can make it.
All of the following facts are true:
These facts are also true:
And so is this final fact:
The limit of a sequence isn't necessarily a member of that sequence.
Because of this, the limit of a sequence doesn't necessarily share any properties with the members of that sequence.
Here, you've seen a sequence of curves of length 4, whose limit does not have length 4.
You've also seen a sequence of jagged, right-angled curves whose limit is not jagged or right-angled at all, but smooth.
Here's another example of this:
1, 1/2, 1/3, 1/4, 1/5, ...
All of the numbers in this sequence are positive, but the limit, 0, is not.
This is not a problem. It's not a contradiction. It's just the way it is.
Breathe in. Breathe out. Carry on with whatever you were doing.