Teach me about epsilon delta definition in limit.

[fast arrow]

Can anyone teach me epsilon delta definition, cause I really don't comprehend about it.

(Lim x→a) f(x)=L
∀ε>0, ∃δ>0, s.t. 0 < |x-a| < δ, | f(x)-L | < ε

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[latent spruce]

@fast arrow https://youtu.be/DdtEQk_DHQs?si=Mf5lNC4DCxgT_RWl

$$(\lim_{x\to a} f(x) = L) \longleftrightarrow \forall \varepsilon>0\exists\delta>0(0 < |x-a| < \delta \rightarrow |f(x)-L|<\varepsilon )$$

[fast arrow]

Em

What's epsilon represent

In that limit

[latent spruce]

@fast arrow watch this vid

he will tell you

[tardy pelican]

(xy)²=(yz)²=(xz)²=xyyzxz=-1

[fast arrow]

Em

I need to know the epsilon stand for what?

[wooden hatch]

It's a tolerance

the inequality |a - b| < epsilon , means that the distance between a and b is less than epsilon, it's like a tolerance in engineering

for example

you need to construct a tube with radius 40 units, tolerance is 1

this means that |40 - the actual thing you constructed| < 1

if you constructed a thing that's 39.5

you would get 40 - 39.5 = 0.5 which is less than our chosen tolerance/epsilon, therefore it's what you were searching for

in the formal definition of limit something similar happens

$|f(x) - L| < \epsilon$

[tardy pelican]

Ambilogy

[wooden hatch]

means that the difference between the function evaluated at a point x near the limit, and the limit, is less than some tolerance epsilon

the distance between them is what I mean.

delta is another tolerance but it's between x and a

the formal definition of limit brought to some informality, basically means:

that no matter what tolerance epsilon for the output of the function (no matter how small is it), you can find another tolerance delta for the input of the function that makes it true/consistent

[fast arrow]

Let say the limit is 5

So you say that f(4.99999)L-f(5) is less than epsilon ?

[fast arrow]

I still can't get it, can you visualise it, (sosorry for my low comprehension...

[wooden hatch]

lets say the limit is 5, now you choose an epsilon, any epsilon

let's say the function is

x+2

[fast arrow]

So

4.99999+2 and 5+2 is less than epsilon?

[wooden hatch]

let me define the function then we'll interpret the formal def

no

First define the function: $f(x) = x + 2$ is our function, now we claim that $\lim_{x \to 3} f(x) = L = 5$

[tardy pelican]

Ambilogy

[wooden hatch]

what this means is

in the formal definition

for any epsilon, for example 0.1

we can find a delta, that will make true $|f(x) - 5| < 0.1$

[tardy pelican]

Ambilogy

[wooden hatch]

look at the graph of the function

when x aproaches 3

f(x) aproaches 5

we can choose any epsilon around 5, for example:

[fast arrow]

x+2-5 = x-3
so x<3.1?

[wooden hatch]

0.1 as I said earlier.

[fast arrow]

Ok

[wooden hatch]

nope

look at what happens

if we look at the point where x aproaches 3

in the graph

the y coordinate is 5

[fast arrow]

It's 5

So?

?

[wooden hatch]

yes, the formal definition of limit is we can choose any neighboorhood around y = 5

and yet we can find a neighborhood around x = 3 that can aproximate it better, look at this graph

this is

5 plus or minus epsilon

well we can find a delta around 3, that can give a better aproximation

what the limit says is

no matter how close the red lines are to 5

you can always find some green lines around the x that will give a better aproximation

the green lines around 3 are the delta

[latent spruce]

https://www.desmos.com/calculator/iejhw8zhqd

[latent spruce]

@fast arrow you can use this to play around

[fast arrow]

No

I need to comprehend...

[fast arrow]

So that's epsilon?

[wooden hatch]

whats the highest math you've studied?

[wooden hatch]

epsilon would be the red lines

[fast arrow]

Ok

[wooden hatch]

actually

it would be the number such that when you sum or substract from 5, you get the red lines

[fast arrow]

So epsilon?

The red line is epsilon?

[wooden hatch]

epsilon is an arbitrary number, no matter which one you choose, that generates those red lines

[latent spruce]

epsilon is the distance between the limit and the red line

[fast arrow]

so

[wooden hatch]

and the formal definition is that you can find another number that generates the blue lines that get closer to the five

[latent spruce]

this thing help tho

[wooden hatch]

yes. The problem is that I'm not sure how to convey it to him since he seems to be trying to understand something above his paygrade and would need to solidify earlier concepts

[fast arrow]

Integration

[wooden hatch]

hmm

∀ε>0, ∃δ>0, s.t. 0 < |x-a| < δ, | f(x)-L | < ε

[fast arrow]

For delta it's also same right?

[wooden hatch]

look at the definition

[fast arrow]

But

[wooden hatch]

translate it into english "for any epsilon, no matter how small, but bigger than 0, you can find a delta bigger than zero such that, if delta is around a, then epsilon is around L"

[fast arrow]

Why the 0< exist in 0<|x-a|<delta for Lim(x→a)?

[latent spruce]

distance cant be 0

[wooden hatch]

yeah

[fast arrow]

Ok

[latent spruce]

because that point in limit doesn't exist

[fast arrow]

Ok im done

[wooden hatch]

you got it or you decided to leave it for later?

[latent spruce]

(remember that limit is approach not equal to)

[fast arrow]

Leave it for later

[wooden hatch]

okay

work a lot with inequalities and absolute values

first

then try again

when you don't understand something, chances are it isn't the day for it or you lack some fundamental concept, therefore work in fundamental concepts, try another day

[latent spruce]

^

[wooden hatch]

you might actually want to revise your informal definition of limit as well, since what might be happening is

that your informal definition is so different from the formal that your brain can't connect what you knew to the new definition

you might've tought that the limit was just evaluating the function at that point but saying it in a fancy way, but suddently you can't actually evaluate the function at that point so it's very hard to make peace with what you thought before

[fast arrow]

Em my friend also learn about function even though he's learning lower math

[wooden hatch]

comparing yourself to other people will demotivate you

compare yoursefl to yourself

are you getting better at math as you study? if yes then good

[fast arrow]

How can I see my post after I close the thread?

[wooden hatch]

no idea

[fast arrow]

+close