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"reach the limits" is correct and usable in written English.
You can use it when referring to something reaching its maximum potential or capacity, or when referring to a physical or figurative limit. For example, "We must be careful not to reach the limits of our budget."
Exact(46)
But they quickly reach the limits of what they can contribute.
I had watched her reach the limits of medicine; her disease was incurable.
The performance of high-end routers will soon reach the limits of conventional technology.
We want to reach the limits of your capacity to digest content and go well beyond them.
As MOSFETs reach the limits of scaling, a variety of physical and application-related constraints must be considered.
Rigidly coupled production lines, as often found in mass production, reach the limits of their flexibility when facing these challenges.
Similar(14)
We will reach the limit".
At sunset, they reach the limit of their map and, apparently, of the world.
After you reach the limit, the money goes in and doesn't count for anything.
Stop when you reach the limit.
As you say, the couples seem to come together, to "stick" until they reach the limit.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com