The actor category has (étale) groupoids as objects and particular left actions, called actors, as morphisms. The first obstacle to computing limits in this category is that the morphisms are not just functions between the underlying groupoids. Pointless topology (and its noncommutative generalisations) provides an avenue of attack here: to any étale groupoid we can associate a canonical inverse semigroup of bisections, encoding both algebraic and topological information. Actors induce semigroup homomorphisms between these bisection semigroups, and under mild topological hypotheses, a groupoid may be recovered completely from the semigroup and lattice structure of its bisection semigroup. Thus, to compute a limit of étale groupoids (satisfying a mild topological condition), one may compute the corresponding limit in the category of semigroups, then reconstruct the desired limit groupoid.