How does the vertical asymptotes affect limits at infinity?

Question

Christopher Allers · Accepted Answer

Vertical asymptotes and limits at infinity do not affect each other. Being a bit careless (but clearer to many than being too careful):
,A vertical asymptote occurs at  #a# if #a# is a finite value for #x# that makes #y# become #oo# or #-oo#. When we ask about the limit at infinity, we are asking if there is a finite values #b# that #y# approaches as x becomes #oo# or #-oo#. The roles of #x# and #y# are reversed.  Which one are we 'trying to make infinite'? Be careful to distinguish the phrases "infinite limit" (#yrarr+-oo#)
as opposed to "limit at infinity (#xrarr+-oo#). Here are 2 links.  (I haven't yet learned to embed them nicely.) Infinite Limits and Vertical Asymptotes:  https://tutor.hix.ai Limits at Infinity and Horizontal Asymptotes:
https://tutor.hix.ai

Axel Alvarez · Answer

undefined Vertical asymptotes do not affect limits at infinity. The behavior of a function near a vertical asymptote is independent of the limit as x approaches positive or negative infinity. The presence of a vertical asymptote only indicates that the function approaches infinity or negative infinity as x approaches a certain value.