# Epsilon delta definition [duplicate]

I understand the definition of limit. I can use it to prove things, but I still don't get the motivation behind it.

• 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. Jul 10, 2019 at 9:25
• "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. Jul 10, 2019 at 17:32

## 1 Answer

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 $$\epsilon$$.