$(\mathrm{\infty },n)$-Limits I: Definition and first consistency results

We give a model-independent definition of limits for diagrams valued in an $(\mathrm{\infty },n)$-category. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of $(\mathrm{\infty },1)$-limits for $(\mathrm{\infty },1)$-categories, and with itself across different values of $n$.