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If I have a limit that's $\lim\limits_{x\to 2}\frac{(x^2-4x+4)}{(x^2-4)} $ which I've factorised and then simplified to $\lim\limits_{x\to 2}\frac{(x-2)}{(x+2)} $ which then has the value of zero, what does that mean exactly? The questions asks me to explore if there exists a limit and what's its value is. If I draw the original expression in a graph it does actually have f(2)=0. Does that mean no limit exists? I'm sorry I'm very confused and don't study maths in English so if I have used the wrong definitions, please be forgiving.

user577215664
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Tryingandstruggling
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The limit is a concept developed in order to deal with the inaccuracies of the concept of an "infinitesimal", an infinitely small quantity, which was used by Newton and Leibniz.

The definition of a limit, in the epsilon-delta sense, encapsulates what it means to approach something (in our case, a real number) a closely as one would wish.

The meaning of a limit which is equal to $0$ is quite simple; You are looking at the function $f(x)=\frac{x-2}{x+2}$, and you want to understand whether the output approaches any value as the input gets closer to $2$. In your case, it does, and that value is $0$.

An interesting question now will be: what is the meaning of a limit the does not exist? to give an example, say you had $f(x)=\frac{x+2}{x-2}$, and you were looking at the same limit. That's a zero in the denominator. Hmm.. could you think of any other situations where the limit would not exist? How would the epsilon-delta definition explain the behavior of a function at a point then?

  • So, the limit doesn't exist as it's undefined? – Tryingandstruggling May 16 '20 at 13:53
  • Actually, not always. In this particular case, yes - there are explicit criteria which determine that. There cases however, where the limit is undefined - but exists, and in such cases one must bring more ideas that will help him show that. The limit concept is quite nuanced. –  May 16 '20 at 14:15