Compute the uncomputable ... sort of
05AB1E, 25 24 bytes
LεÅγιøIãʒηεøJË}à}gĀ}'.šJ

Try it online!
-1 thanks to @ovs
f(n) is a decimal number such that it's xth digit is 1 iff x&lt;=n and when you run length encode x and convert it to matching pairs, it has a solution to PCP with a length of at most n pairs.
outputs as .digits after the dot. In case the 0 is needed, it's +1 to replace '. with „0.
Code explaination:
L      all numbers from 1 to N
ε      map each to
 Åγ    run length encode
 ι     uninterleave to two lists
 ø     tranpose
 Iã    cartesian power N, all choices of N such pairs
 ʒ     only keep those such that
  η    its prefixes
  ε    map each to
   ø   itself transposed
   J   joined
   Ë   both elements are equal
  }
  à    maximum - if any of those are 1
 }
 gĀ    the length of the filtered list isn't 0
}
'.š    prepend .
J      join, and implicitly print

We can show a reduction from PCP with two symbols (which is known to be undecidable):
given an instance of PCP with two symbols, 1 and 2 (for example [2,1], [11,2], [2212,111]), unflatten the pairs (which would give [2,1,11,2,2212,111] in the example) and run length decode with alternating 1, 0. That number would be x. The solution to the PCP instance is 1 iff the xth digit of the limit of this function is 1.