Terminology - Limit doesn't exist

Question

Take the following limit: $$ \lim_{x \to 2} \dfrac{x+2}{x-2} $$

This doesn't exist. My textbook says it doesn't because "The denominator approaches 0 (from both sides) while the numerator does not."

I don't understand what this means. I do understand that it doesn't exist. My thought process is that the left limit and the right limit aren't equal, so the limit doesn't exist.

But I want to know what is meant by the text in the textbook. Can anybody give me an example where the numerator also approaches 0 from both sides?

Answer 1

Consider the function $\frac{\sin x}{x}$ as x approaches 0 for a case where the limit does exist and is equal to 1. Reference if you want one for this case.

If you want another example where the limit doesn't exist consider either $\sin x$ as x tends to infinity or $(-1)^n$ as n tends to infinity, consider the cases where n is a sequence of odd numbers getting bigger versus n being even numbers getting bigger if you want more of an explanation, where because each is periodic, there are ways to create a contradiction if there was a value.

Answer 2

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.