Calculus Limit finding threshold number

[spice ibex]

I have learned this in school, now in uni they are teaching it with "guessing" and I don't have my notebooks currently and can't see how this should be solved without "guessing". Can someone remind me the way? I have been searching on the internet the last 30 minutes, can't find any of these examples especially.

[candid birch]

(3n^2 - 2n)/(4n^2 - 5)
= (3n^2 - 2n)/(4n^2 - 5) * (1/n^2)/(1/n^2)
= (3 - 2/n)/(4 - 5/n^2)```

Then as n goes to infinity, 2/n and 5/n^2 go to 0.

[spice ibex]

so I get 3/4, but I have to get this:

[candid birch]

...wait, are you asking to be reminded of the epsilon-delta definition of a limit?

[spice ibex]

is this that?

[candid birch]

Well, you have an epsilon.

[spice ibex]

I looked into that, non of it looked like this

[candid birch]

That's because the epsilon-delta form specifically is for finite limits of finite values; that is, if we're talking about lim_(x -> a) f(x) = L, both a and L have to be finite for the epsilon-delta form to apply.

[spice ibex]

so I can't apply it here?

since a is infinite

So the only way solving this, is "guessing"?

*estimating

@candid birch ?

[candid birch]

No.

lim_(x -> inf) f(x) = L if and only if, for all e > 0, there exists some N such that N < x implies |f(x) - L| < e.

[spice ibex]

Yeah, I can get to |f(x) - L| < e but what to do after that? I have different quotients

((8n - 15) / (16x^2 - 20)) < e

((8n - 15) / ((4n - 2sqrt(5)) * (4n + 2sqrt(5))) < e

[candid birch]

I'm not actually sure.

[spice ibex]

Can you tell me what is this problem called?

[candid birch]

I don't know what precisely you mean by "this problem", so no.

[spice ibex]

Finding the threshold number, known the limit and the equation

[candid birch]

That... doesn't really mean much to me.

If you're just looking for something to Google, "epsilon n limit" would be a good place to start.

[spice ibex]

Finite Limit at Infinity

[candid birch]

...yes, which is an epsilon n limit.

[spice ibex]

oh yeaaah, finally found stuff, THANKS ❤️ Sadly they do the "guessing" 😦 So wasted 2 hours getting back where I started. https://www.youtube.com/watch?v=bAtbCHcm8gM

Thanks ❤️ @candid birch

[candid birch]

What precisely do you mean by "guessing"?

[spice ibex]

removing the 10

[candid birch]

That's not really guessing.

[spice ibex]

estimating

[candid birch]

Still no.

[spice ibex]

I don't feel like thats a mathematical step

[candid birch]

They're not saying that 11/(4n + 10) = 11/4n. They're saying that 11/(4n + 10) < 11/4n, which is true.

[spice ibex]

but you can say 11/(4n + 10) < 11/(0.1n)

[candid birch]

You totally can.

[spice ibex]

so there are endless solutions basically

[candid birch]

Well, yeah. Because there's a lower bound that N must be greater than, but N can be greater than anything greater than that.

But we don't care what N is, just that it exists for all e.

[spice ibex]

How do you keep in your mind such a lot informations?

[candid birch]

Practice.

[spice ibex]

Im sure I did atleast 20 of these in school a year ago, now I have totally forgotten everything, not just this. Do you practice a lot of stuff on a daily basis?

[candid birch]

Not really. It's more like you practice until it clicks, and then you're good.

[spice ibex]

Guess I don't have a mouse to click. Anyway, thanks for the help ❤️