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### Is the choice of epsilon delta free in limit definition? [duplicate]

We use limit definition as: $$\lim_{x \to a} f(x)=L$$ if $$\forall \varepsilon > 0 \quad \exists \delta > 0 \quad (|x−a| < \delta \implies |f(x)−L| < \varepsilon)$$ But why we ...
507 views

### Why does $\epsilon$ come first in the $\epsilon-\delta$ definition of limit? [duplicate]

As we know, $\underset{x\rightarrow c}{\lim}f(x)=L\Leftrightarrow$ for every $\epsilon>0$ there exists $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$. My question ...
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### Can the definition of continuity be said both of these ways? [duplicate]

So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because ...
563 views

### Switching the definition of epsilon-delta limit [duplicate]

What if the epsilon-delta definition of a limit reversed the wording for $δ$ and $ϵ$: “for all $δ>0$, there exists an $ϵ>0$ such that, if $0<|x-a|<δ$, then $|f(x)-L|<ϵ$.” Would this ...
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### Why is the definition of “limit” difficult to understand at first?

Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
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### Why can't epsilon depend on delta instead?

When presented with $\lim_{x\to a}f(x) = L$, we are usually taught to intuitively think of $x$ approaching the value $a$ from both sides, with $f(x)$ getting closer and closer to the value $L$. For ...
WHY are we always given $\epsilon > 0$ first, then solving for a $\delta>0$? This is in the limit definition. I want to ask: Can we say "given $\delta>0$, there exists $\epsilon>0$"? ...