Obviously it depends on the definition of "exists". Some authors explicitly work over the extended real line with $\pm\infty$ adjoined, so that such infinite limits do explicitly "exist" as first-class values. But there is no consensus. One needs to pay attention to the author's definitions and conventions.
Perhaps it is worth mention - even though this case is rather trivial - that adjoining points at infinity is a special case of various constructions that attempt to simplify matters by some type of existential closure. Below I append an excerpt from an old 1996 post where I give a little further discussion:
This thread originated in a query as to whether infinity or 1/0
could be admitted as a "value", and soon drifted into discussion
of the Riemann sphere and other topological manifestations of
infinity via compactification. Below I point out a couple of
marvelous references on these topics; further I would like to
bring to your attention a much wider perspective on such topics,
namely that of existential closure as studied in model theory.
There is a beautiful exposition of points at infinity, projective
closure, compactifications, modifications, etc. in [FM] Chapter 7,
Points at Infinity, by H. Behnke and H. Grauert. This is volume III
in the excellent "Fundamentals of Mathematics" series,
which deserves to be on the bookshelf of every budding mathematician.
A much deeper appreciation of the methodology behind these constructions
can be had by studying them from a model-theoretic perspective, in
particular from the standpoint of existential closure and model
completion. Kenneth Manders has written a series of thought
provoking papers , from this perspective.
I close with an excerpt from the introduction to :
"The systematic adjunction of roots, or solutions to other simple
conditions, as in formation of the complex numbers by adjoining
imaginaries, or in adjunction of points "at infinity" in traditional
geometry, may be analysed as existential closure and model
completion. 'Existential closure' refers to a class of processes
which attempt to round off a domain and simplify its theory by
adjoining elements -- more properly, it refers to the formal
relationship that obtains in such a process. 'Model completion' is
one of the terms employed when this process is successful. The
formation of the complex numbers, and the move from affine to
projective geometry, are successes of this kind. Thus, the theory of
existential closure gives a theoretical basis of Hilbert's "method
of ideal elements." I now sketch the theory of existential closure,
to bring out when, how, and in what sense existential closure gives
[FM] Fundamentals of mathematics. Vol. III. Analysis.
Edited by H. Behnke, F. Bachmann, K. Fladt and W. Suss.
Translated from the second German edition by S. H. Gould.
Reprint of the 1974 edition. MIT Press,
Cambridge, Mass.-London, 1983. xiii+541 pp. ISBN: 0-262-52095-8 00A05
 Manders, Kenneth
Domain extension and the philosophy of mathematics.
J. Philos. 86 (1989), no. 10, 553--562.
 Manders, Kenneth L.
Logic and conceptual relationships in mathematics.
Logic colloquium '85 (Orsay, 1985), 193--211,
Stud. Logic Found. Math., 122,
North-Holland, Amsterdam-New York, 1987.