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We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $({ℒ}_i,{\alpha }_{{ℒ}_i})$ is (isomorphic to) the direct limit of universal central extensions of $({ℒ}_i,{\alpha }_{{ℒ}_i})$. As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras ${\left\{({\mathrm{sl}}_k\left(å\right),{\alpha }_k)\right\}}_{k\in I}$ and describe the universal central extension of its direct limit.