  We prove a series of Approximation Theorems in the setting of Waldhausen
quasicategories. These theorems, inspired by Waldhausen's 1985 Approximation
Theorem, give sufficient conditions for an exact functor of Waldhausen
quasicategories to induce a level-wise weak homotopy equivalence of K-theory
spectra.
  The Pre-Approximation Theorem, which holds in the general setting of
quasicategories without Waldhausen structures, provides sufficient conditions
for a functor F:A->B to restrict to an equivalence of the maximal
infinity-groupoids in A and B.
  Our Approximation Theorems follow from the Pre-Approximation Theorem. The
Approximation Theorem in the quasicategorical setting most analogous to
Waldhausen's is: if an exact functor F:A -> B satisfies Waldhausen's App 1 and
App 2, and the domain A admits colimits of the aforementioned type and F
preserves them, then K(F) is a level-wise equivalence. As a corollary, if F is
an exact functor with ho(F) an equivalence of ordinary categories, and every
morphism in the domain A is a cofibration, then K(F) is a level-wise
equivalence.
  We then introduce a version of App 2 called Cofibration App 2 that only
requires factorization of cofibrations Fa >-> b as (equiv) o F(cofibration) and
prove an analogous Cofibration Approximation Theorem, and a corollary for
certain functors that induce an equivalence of cofibration homotopy categories.
  We also prove that S_n^infinity is Waldhausen equivalent to
\overline{\mathcal{F}_{n-1}^infinity} using the mid anodyne maps known as spine
inclusions, and clarify how hypotheses and notions in Waldhausen structures are
related in new ways in the context of quasicategories.
