You are correct: the limit doesn't exist because the left limit (negative infinity) and the right limit (positive infinity) are not equal.
However, there are people, especially teachers of beginning students, who treat all instances of "limit is infinity" (which "denominator approaches 0 (from both sides) while the numerator does not" is an example of) as "limit does not exist".
I don't, because I think it obscures an important nuance: that vertical asymptotes behave more like removable discontinuities (holes) than they do jump discontinuities. Sometimes, I even go so far as to equate positive infinity and negative infinity, using what is known as the projective real line - symbolized $\mathbb{R}^*$. (This is probably why your textbook specifies the "limit is infinity" version of "limit does not exist".)
But I also understand that the nuance may be too subtle for beginning students, and treating infinity as a number brings problems of its own. Namely: What is the inverse of infinity? How does it interact with addition and multiplication? How does it interact with $0$? These are questions that must be answered and cannot be answered simply.
So teachers and textbooks tend to use one of three approaches: lump infinity in with "does not exist", split the two cases but call "$\lim = \infty$" an abuse of notation (or using $\lim \rightarrow \infty$ instead), or go into a discussion about the projective real line (with or without calling it the projective real line) and risk derailing the class.