I understand the definition of limit. I can use it to prove things, but I still don't get the motivation behind it.
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1$\begingroup$ If you can get a copy of Spivak's calculus book, I suggest you take a look at chapter $5$, where he explains the definition of limits. He guides the reader in getting from the intuitive meaning of limits to the rigorous definition of limits in a step-by-step manner. $\endgroup$– peek-a-booJul 10, 2019 at 9:25
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$\begingroup$ "Why isn't there a definition even after 150 years that corresponds better with the intuitive idea of limit?" The standard definition corresponds very well with my intuitive notion of a limit. $\endgroup$– David C. UllrichJul 10, 2019 at 17:32
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As someone who took intro real analysis last quarter, I can relate.
"definition is somehow not capturing the intuitive idea of limit as a variable moving towards a value."
It does, I just think you need to think through the definition more, perhaps see some visuals. The key is word is $\forall \epsilon > 0$. Imagine starting from a large $\epsilon$ value. You can then find a $\delta$ in the domain for which the definition holds. Now imagine shrinking $\epsilon$ by a little, you can find another $\delta$ which works. Keep shrinking $\epsilon$ until it is super small. You can still find a $\delta$ and a corresponding region for which the definition holds.
This visual might help. https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit . Just imagine shrinking the $\epsilon$.
