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I understand the idea behind it and I've even seen an excellent visual representation of it from 3Blue1Brown, but I have no idea how to use it in a practical sense. How do I use the formal definition to prove the following limit?

$$\lim_{x \rightarrow 2}{\frac{x-2}{x^2-2x}}$$

How do I use it to prove that this limit doesn't exist:

$$\lim_{x \rightarrow 0}{\frac{1}{x}}$$

How about one that doesn't go towards infinity but still doesn't exist:

$$\lim_{x \rightarrow 0}{\frac{|x|}{x}}$$

I can prove all of these easily by just doing it the "regular" way through some algebraic manipulation followed by applying the limit, or by just using logical reasoning. But there seems to be this notion that this isn't the "formal" way of doing it. Why?

ShootinLemons
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    Can you please define the limit without using the $\epsilon$-$\delta$ definition? – mathcounterexamples.net Jan 28 '22 at 19:55
  • As you mentioned, the definition of limit involves delta-epsilon so just doing formal algebraic manipulations will not help to prove that the obtained number is indeed the limit. – Vasili Jan 28 '22 at 19:57
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    This post might be of some use to you for the first problem https://math.stackexchange.com/questions/418961/epsilon-delta-proof-that-lim-limits-x-to-1-frac1x-1 – Irving Rabin Jan 28 '22 at 20:46
  • For the second limit, how would you prove that there is some real number $,x,$ such that $,|x|>100?$ What about $,|x|>1000?$ And so on. – Somos Jan 28 '22 at 22:44
  • Think about $ε$ and $δ$ as some parameters which you can control like if $x < ε$ then as $ε$ goes to zero x also goes to zero also check out professor leonard on YouTube he has a great playlist on calculus 1 2 and 3 (I will link it sometime later if you cannot find it, just ping me) – Sam Hoffman Jan 30 '22 at 09:55

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