explain epsilon-delta definition of a limit

[Anonymous No.16139011]

i don't get it,explain it to me please
i'm a high school student btw

[Anonymous No.16139024]

|x-a| is the magnitude of distance between an x value and the value a that it's "going to" in the limit
|f(x)-L| is the magnitude of distance between what the function f spits out, and the value you're trying to prove the limit goes to
The idea is that if x stays within a certain tolerance that makes it sufficiently close to a then the same holds for f(x) and L, so it's "closing in" on those values within the tolerances delta and epsilon

it is a logic statement "if X then Y" so to prove that you need to assume X is true (otherwise it's vacuous) and then show that Y is true.

[Anonymous No.16139096]

>>16139024
i still don't completely get it but thank you

[Anonymous No.16139170]

you have to kinda read it backwards (blame jewish polish notation for modern analysis)

1. the goal is to show that a function f(x) has a limit L at the point x=a
2. this means that if i take an epsilon band around L , so (L-eps , L + eps) , we should find a delta (possibly depending on eps) that gives us a 'delta band' (a-del , a+del) which guarantees that any input x taken from the delta band gives an output f(x) in the epsilon band.

so the steps are:
1. write down |f(x) - L | < eps
2. find a delta d(eps) that defines your input band |x-a| < delta(eps)
3. simplify |f(x) -L| in such a way as to extract (x-a) or something like that so you can bound it by a function of delta(eps)
4. write the bound in terms of epsilon , for example if you end up with (blah) < 10delta
and delta(eps) = 2*eps , then this is just (blah) < 20*eps
5. write down the steps but in reverse: so you write given eps >0 , let delta = 2*eps , then if |x-a| < delta , then |f(x) - L| < eps

[Anonymous No.16139874]

>>16139011
read an analysis book like Amann Escher. It'll make sense once you get to it

[Anonymous No.16140005]

>>16139011
You already made a thread about this
>>/sci/thread/15922473
We say the limit of f at a number x0 is y0 if you can make the image of f(x) arbitrarily close to a number y0 (within +-ε of y0) by making x "close enough" to a number x0 (within +-δ of y0).
What's so hard to understand about that?

[Anonymous No.16140045]

>>16139170
wtf is the point of any of that? besides asymptotic functions and fake crap like the step function, what wouldn't have those properties?

[Anonymous No.16140047]

>It takes a faculty supplanting to
Borderline technicity intelligence
>Neck yourself, faggot

[Anonymous No.16140066]

Ask someone to invent smallest number epsilon they can say.

You are prepared. For every possible epsilon, you have figured a number delta. Around domain point 'a', you have drawn a circle with radius delta. You know every value of them will be mapped inside a circle around L, with radius epsilon, in the function value space.

[Anonymous No.16140109]

underage b&

[Anonymous No.16140185]

It says that for any distance, epsilon on the y axis btn points in your function, you can choose or find a corresponding distance delta, on the x axis, such that the epsilon is not unbounded. Take any distance on the y axis, and track the point on your curve on the x axis, you will form a right angled triangle or a horizontal line that is not too big,if you can't do this, then your function doesn't have a limit at that point x, i.e curves that have asymptotes like tan x or 1/x and therefore no limit at x=90 and x=0 respectively

[Anonymous No.16140212]

>>16140185
thank you , i think i get it now
>>16140005
>You already made a thread about this
no i didn't

[Anonymous No.16140226]

>>16139011
Read Calculus by Tarasov

[Anonymous No.16140993]

>>16139011
The goal of ε-δ definition is to tackle infinity in a meaningful way. This analogy is not mine but think of it like this: remember when you were a kid and you were in a contest to say the biggest number? Probably one of you said at one point "whatever number you say + 1"

The definition basically says for whatever points within ε-tolerance of L, there is a corresponding range of points within δ-tolerance of a. The goal is to find δ in terms of ε. Why? because of our analogy above. we made epsilon arbitrary, and so we could choose it to be as close as we would like to the point L. In the example I draw, the limit as x tends to 1 doesn't exist. I literally can't make ε tolerance as small as I'd like. I cannot make it smaller than 1. This is why It doesn't exist. I suggest looking at cases where things fail to have a limit, it makes you understand it better.