# Questions tagged [epsilon-delta]

For questions regarding $\varepsilon$-$\delta$ definitions of limits, continuity of functions and $\varepsilon-N$ definition of limit of sequences.

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### Alternate epsilon-delta proof of the limit of the reciprocal function

I'm trying to prove the following using the epsilon-delta method: $$\lim_{x\to 1}\frac{1}{x} = 1$$ Prior to reading the fantastic answer by @Clarinetist in this question, I tried writing the proof ...
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### Find the limit and give the strict $\epsilon - \delta$ proof that: $\lim_{x\to 1^{+}}$ $x\lfloor x \rfloor$

This is from Advance Calculus Theory and Practice By John S. Petrovic. The book has the answer in the back but I need some clarity on $\delta$. Answer: The limit is 1. Let $\epsilon > 0$, and ...
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### Doubt about the Epsilon-Delta Definition of the limit of a function at $a$ point.

My math book states that for a limit to exist at a point $a$ of a function $f$, $f(a)$, must be equal to the limiting values from each side of the point. However, how does the epsilon-delta definition ...
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### Proof that the extremum of a function is a critical point [duplicate]

I'm following a proof given by Spivak in his textbook "Calculus" and one of the steps seems slightly unjustified to me. Firstly, here's the theorem: If $f$ is a function defined on $(a,b)$ ...
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### Seeking feedback on my $\epsilon$-$\delta$ proof of $\lim_{x \to a} x^2 = a^2$.

I'm seeking feedback on my understanding of the $\epsilon$-$\delta$ limit proof for quadratic functions, specifically for $\lim_{x \to a} x^2 = a^2$. After studying multiple proofs, I've noticed that ...
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1 vote
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### how can I continue from here proving there is no limit to the right of $x_0=0$

when composing g on f at x=0 and checking if there exists A limit to the right of x, I get that there is no limit this is because if I wrote the composition correctly as I take numbers larger than ...
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### Adjacent vs. All-Possible Distances in the Definition of a Cauchy Sequence

I'm currently learning about Cauchy sequences, and I'm trying to build my intuition regarding its definition. My question is on how we can motivate why we consider the distances between all points and ...
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### Can I plug in numbers for x when proving limits?

At what point can I plug in values for x when proving limits? I know I can't have delta be a function of X. I also know delta should be epsilon divided by 9, but am I allowed to just start plugging in ...
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### Using extra inequalities to prove $\epsilon$-$\delta$ continuity questions.

Context: high school student that read some math books is now reading a calculus book (I'll call it G for the sake of repetition). Still didn't make it to the limit definition. So, there is a ...
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### How can any supremum of a proper subset of the real numbers be outside the set, following the logic below?

Let E ⊂ R, E is bounded above, and α = sup E. Then ∀ε > 0, there exists some t ∈ E such that α − ε < t ⇒ α − t < ε Since α > t, α − t = |α − t|. It follows that |α − t| < ε which ...
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### In what ways does the properties of infinity alter or invalidate this epsilon-delta proof?

Disclaimer: I am following a math textbook which has not really covered the properties of infinities yet. Only introduced the concept in the context of limits. I have the following problem in my ...
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### Why is $\lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy}$ defined while $\lim_{(x,y,z) \to (0,0,0)} \frac{\sin(xyz)}{xyz}$ is not?

I've found an exercise that compares the following limits: $$\lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy} \qquad\text{and}\qquad \lim_{(x,y,z) \to (0,0,0)} \frac{\sin(xyz)}{xyz}$$ The solutions ...
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### Followup for proof of scalar multiplication of limits

In a previous question of the $\epsilon$ - $\delta$ proof of the scalar multiplication property of limits, they say that $\delta$ is $\frac{\epsilon}{|c|}$ - but why is $c$ absolute? Sorry, quite ...
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### Structure required for differentiability in topological context

I am a bit unsure about precisely what structure is required for a function to be differentiable, or even continuous for that matter. Specifically, what structure are we using in the $\epsilon-\delta$ ...
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