  We initiate a study of punctured tubular neighborhoods and homotopy theory at
infinity in motivic settings. We use the six functors formalism to give an
intrinsic definition of the stable motivic homotopy type at infinity of an
algebraic variety. Our main computational tools include cdh-descent for normal
crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under
$\ell$-adic realization, the motive at infinity recovers a formula for
vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's
limiting Hodge structures and Wildeshaus' boundary motives. Under the
topological Betti realization, the stable motivic homotopy type at infinity of
an algebraic variety recovers the singular complex at infinity of the
corresponding topological space. We coin the notion of homotopically smooth
morphisms with respect to a motivic $\infty$-category and use it to show a
generalization to virtual vector bundles of Morel-Voevodsky's purity theorem,
which yields an escalated form of Atiyah duality with compact support. Further,
we study a quadratic refinement of intersection degrees, taking values in
motivic cohomotopy groups. For relative surfaces, we show the stable motivic
homotopy type at infinity witnesses a quadratic version of Mumford's plumbing
construction for smooth complex algebraic surfaces. Our construction and
computation of stable motivic links of Du Val singularities on normal surfaces
is expressed entirely in terms of Dynkin diagrams. In characteristic p>0, this
improves Artin's analysis on Du Val singularities through \'etale local
fundamental groups. The main results in the paper are also valid for
$\ell$-adic sheaves, mixed Hodge modules, and more generally motivic
$\infty$-categories.
