If we model our (∞,1)-categories by quasicategories, then this can be made precise by saying it is a limit over some simplicial set with finitely many nondegenerate simplices. Note that such a simplicial set is rarely itself a quasicategory; we regard it instead as a finite presentation of a quasicategory.
This appears as (Lurie, cor. 18.104.22.168).
This appears as (Lurie, prop. 22.214.171.124).
Binary products, pullbacks, and terminal objects are all finite -limits.