Linked Questions

2
votes
5answers
655 views

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 ...
6
votes
5answers
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 ...
2
votes
4answers
152 views

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 ...
2
votes
1answer
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 ...
21
votes
5answers
7k views

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 ...
11
votes
3answers
3k views

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 ...
3
votes
8answers
488 views

In the epsilon-delta definition, what is wrong if I said: “given delta, there exists an epsilon”?

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 $\epsilon>0$"? ...
1
vote
3answers
945 views

Why does $U(f,P) - L(f,P) < \epsilon$ make a good criterion for integrability?

That is $f$ is integrable on $[a,b] \iff \forall\epsilon>0, \exists P$ of $[a,b]$ such that $$U(f,P) - L(f,P) < \epsilon$$ I was thinking that a better definition would be if $U(f,P) =...
3
votes
5answers
416 views

When $\delta$ decreases should $\epsilon$ decrease? (In the definition of a limit when x approaches $a$ should $f(x)$ approach its limit $L$? )

Assume that the function $f$ has the following limit: $$\lim_{x \rightarrow a} f(x) = c$$ if I have the correct understanding of what this definition should mean, the following should be true: $$ \...