Yes, they do. Looking at it, it's not really obvious why, though. The rule (to which I can think of some exceptions) is that we use of with the, and to with a. Like this:
There is a limit to his endurance.
He has reached the limit of his endurance.
I can't think of a circumstance in which you would go wrong using this rule. Still, you might see something like this:
The limit to what I'm willing to spend is $100.
The reason that you might see this is that it could imply this:
There is a limit to what I'm willing to spend. The limit to what I'm willing to spend is $100.
So, if you are talking about "the limit" in reference to "a limit" that has been stated or implied, then you might use to instead of of.
Now, here you would generally use of:
$100 is the limit of what I'm willing to spend.
So, while there might be some exceptions to the rule, you can still apply it in those situations without going wrong.