What is the clue to solve delta-epsilon proofs?

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What is the clue to solve delta-epsilon proofs?

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Pick any epsilon, find delta

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>>7846590

Get gud with inequalities

The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by Steele

Inequalities by Hardy, Littlewood, and Polya

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>>7846590

Do you have an example that you'd like to work through?

General methodology:

- Work backwards, start with the inequality bounded by epsilon that you want to obtain and try to work back towards being bounded by delta, then choose that delta.

- Leave gaps in your work and jump to what you would like to achieve. Sometimes the delta to pick becomes obvious that way.

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>>7846624

I was working on this one, buy I couldn't solve it.

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>>7846652

Please give the definition of convergence that you are working with too.

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>>7846670

I don't understand what that means

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>>7846652

Let [math] \varepsilon > 0[/math] choose [math] \delta [/math] such that [math] \frac{1}{4 \delta^{-1} + 2 } < \varepsilon [/math] then for all [math] x \in (2 - \delta, 2 + \delta) [/math] we have

[eqn] \left| \frac{1}{x} - \frac{1}{2} \right| = |2 - x| \left| \frac{1}{2x} \right| < \delta \frac{1}{2 (2 + \delta)} < \varepsilon [/eqn]

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>>7846684

How are you trying to prove that this is true using epsilon-delta?

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holy shit, fuck analysis, I can't believe I got out of that class unscathed

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>>7846705

Good job buddy, your later modules will build on it and generalise it.

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bee urslef

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>>7846695

By proving this

>>7846694

I got to the same result, but I couldn't define delta in terms of epsilon.

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>>7846724

And how would you prove that a general function f from X to the reals converges to c?

I'm asking you to state the base definition of what it means for a function to converge (which you are using).

>I got to the same result, but I couldn't define delta in terms of epsilon.

I'm not that anon, but you're looking for motivation/clues on how to tell which delta to pick right?

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Someone do delta-epsilon proof for limx->0 sinx/x = 1

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>>7846736

He means your basic calc 1 definition of the limit autistic faggot.

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>>7846652

Haven't taken an analysis class but know the basics of an epsilon-delta proof, could you use the fact that [eqn]\lim\limits_{x \to c}\frac{f(x)}{g(x)}=\frac{\lim\limits_{x \to c}f(x)}{\lim\limits_{x \to c}g(x)}[/eqn] to say something about how [eqn]\lim\limits_{x \to 2}\dfrac1x=\frac{\lim\limits_{x \to 2}1}{\lim\limits_{x \to 2}x}[/eqn]and just go from there?

Sorry my Tex is so shitty btw this is literally my first time ever using it.

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>>7847431

Fuck, it shows up fine in the preview but I don't doubt I typed something in horribly wrong anyway.

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>>7846594

kek

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git gud with logical quantifiers. If you're a pro at formal logic then this stuff becomes a brainless mechanical process.

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>>7847414

What's calc 1?

And there are slight variations in the definition from author to author, which is why it's good practice to get someone to first state it if they're struggling with it.

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>>7847408

This is trivial once you recall that sin(x) < x < tan(x) for x in(0, pi/2).

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>>7847689

>recall that sin(x) < x < tan(x) for x in(0, pi/2).

This would be circular reasoning because you need to know that the derivative of sine is cosine to prove this.

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>>7847730

Actually no! We can resort to geometrical constructions to prove this.

Would you like to attempt this for yourself before I give a proof?

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>>7846590

It's all about finding bounds.

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>>7847408

You need an algebraic definition of sin(x) to do that. If your definition is geometric, the proof is also geometric.

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>>7847686

Calc one starts with limits and usually ends with very basic integration (u-substitution, or riemann integrals)

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How important is a good grasp of epsilon-delta proofs later on?

For some reason it's the only thing that I just can't seem to get and gives me anxiety. I don't want to use epsilon-delta proofs ever.

At how much of a disadvantage am I if I just let it be and don't learn to deal with epsilon-delta?

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isn't epislon-delta just a proof by exhaustion for the existence of infinitesimals?

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You need to find a function [math]g: mathbb{R} \leftarrow mathbb{R}[/math] that maps one number [math]\delta[/math] for every [math]\varepsilon[/math] such that the delta-epsilon condition (x distance is smaller than delta and greater than zero; f(x) distance is smaller than epsilon) is satisfied.

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>>7849209

So then what is analysis for you guys?

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>>7846590

git gud

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>>7849447

Epsilon-delta is considered baby stuff that everyone knows. The methods of bounding that you should get from it gets used later on, so you really should learn it. When you think of it, perhaps you should draw a picture on paper or in your mind.

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>>7846705

You guys only see it on analysis?

I had to make proofs on Calculus 1.Is it the norm in other countries? or was I just fucked? I feel like all my calculus classes had a little too much demonstrations, even though i'm EE. (supposedly more focused on problem-solving)

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>>7847737

Since no one bothered to attempt the exercise, proof is attached in the picture.

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>>7849993

all of that done formally w/ epsilon delta

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>>7850008

Now to solve >>7847408

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>>7850016

thank you very much!

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>>7846724

Note that,

$$ \frac{\delta}{2(2+\delta)} < \frac{\delta}{4}, $$

so we can choose $\delta = 4\epsilon$.

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>>7850312

>Note that,

>[eqn]\frac{\delta}{2(2+\delta)} < \frac{\delta}{4}, [/eqn]

>so we can choose [math]\delta = 4\epsilon[\math].

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>>7850321

You fucked up.

Let [math] \varepsilon = 100, \delta = 400[/math] and [math]x = \frac{1}{1000}[/math] then

[math]|x - 2| < \delta [/math] but [math] \left| \frac{1}{x} - \frac{1}{2} \right| > \varepsilon [/math]

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