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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?

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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.

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