Question

How does the vertical asymptotes affect limits at infinity?

Answer 1

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: http://socratic.org/calculus/limits/infinite-limits-and-vertical-asymptotes

Limits at Infinity and Horizontal Asymptotes:
http://socratic.org/calculus/limits/limits-at-infinity-and-horizontal-asymptotes