Hi /sci/, so about three days ago I made a thread asking people

>>9133582
http://boards.4chan.org/sci/archive
https://warosu.org/sci/
Put in some keywords in the second.

>>9133582
dude this is so wrong

keep in mind that you want to show that the expression is similar to t^2 / t^3 or 1 / t for sufficiently large t
dont be afraid of using words to explain what you are doing in your working

no fucking idea what happened after |(t+2)/((t-1)(t-3))| desu

>>9133582
Do you need to use a direct epsilon delta argument or are you able to use theorems? Because with just the squeeze theorem you can do this without much effort and just conclude with an epsilon delta argument.

>>9133582
This isn't actually very hard right? I'm a physicist so my way of thinking may not be absolutely rigorous:
working out the brackets yields a term of [math] t^3 [math/]. If you then divide above and below the denominator by [math] t^2 [math/] the term above will go to 1, and the term below is simply t, as the [math] \frac{t^3}{t^2} [math/] is t.
now letting t appoach infinity, all our terms with 1/t go to 0, as will our function, since that too is simply 1/t.

>>9133582
Why the fuck do you want to prove that with epsilon
Just solve it like everyone
[math]a_t=\frac{t^2+4}{(t-1)(t+2)(t-3)}\sim\frac{1}{t}\underset{t\to +\infty}{\longrightarrow}0[/math]
And be done with it

>>9134080
You can't do that with a delta epsilon proof.

You have to show that the terms in the sequence eventually tend to some number. You have to start the proof by the logical quantifier. If the first line of this proof isn't
"Assume epsilon greater than 0" it's automatically wrong.

The next and hardest part is finding a tail of the sequence so that you know the sequence converges. It's not clear to me what this is, but you have to start by removing the absolute value bars, which will result in an inequality that you use for N. From there, the proof is straightforward plug and chug.

File: giannopoulos_20-07-16.jpg (2MB, 4032x3024px) Image search: [Google]

2MB, 4032x3024px

I think i've got it:
We have the sequence: [math]a_{t} : \mathbb{N} \setminus {1,3} \longrightarrow \mathbb{R} [/math] .
We want to show that: [math] \forall \epsilon > 0 : \exists N(\epsilon) > 0 : \forall t>N(\epsilon) : |a_{t}|< \epsilon [/math] .

Now,
[math] \left|\frac{t^2+4}{(t-1)(t+2)(t-3)}\right|<\left| \frac{t+2}{(t-1)(t-3)} \right| = \left| \frac{t+2}{t^2-4t+3} \right| [/math] as you showed.
Using partial fractions, you can write it as follows:
[math]\left| \frac{t+2}{t^2-4t+3}\right| = \frac{5}{2} \left| \frac{1}{t-3} \right| - \frac{3}{2} \left| \frac{1}{t-1} \right| [/math] .

Then, let [math] N_{1}(\epsilon):=3+ \lfloor \frac{1}{2 \epsilon} \rfloor \frac{5}{2} [/math]

[math] \forall t \in \mathbb{N} \setminus {1,3}: [/math]
[math]t> N_{1}(\epsilon)=3+ \lfloor \frac{1}{2 \epsilon} \rfloor \frac{5}{2}[/math]
[math] \Rightarrow t-3>\lfloor \frac{1}{2 \epsilon} \rfloor \frac{5}{2} [/math]
[math] \Rightarrow \frac{2}{5} (t-3) > \frac{1}{2 \epsilon} [/math]
[math] \Rightarrow \frac{5}{2} \left| \frac{1}{t-3} \right|<2 \epsilon[/math]

And define: [math] N_{2} (\epsilon) := 1 + \lfloor \frac{1}{ \epsilon} \rfloor \frac{3}{2} [/math]

[math] \forall t \in \mathbb{N} \setminus {1,3}: [/math]
[math]t>N_{2} (\epsilon)=1 + \lfloor \frac{1}{ \epsilon} \rfloor \frac{3}{2}[/math]
[math] \Rightarrow \frac{2}{3}(t-1)>\frac{1}{ \epsilon}[/math]
[math] \Rightarrow \frac{3}{2} \left| \frac{1}{t-1} \right|< \epsilon [/math]

Now, let [math] N( \epsilon) := min{N_{1}( \epsilon),N_{2}( \epsilon)} [/math]

then, [math]\forall t>N( \epsilon) [/math]:

[math]\left| a_{t} \right|<\frac{5}{2} \left| \frac{1}{t-3} \right| - \frac{3}{2} \left| \frac{1}{t-1} \right|<2 \epsilon - \epsilon [/math]
[math] \Rightarrow \left| a_{t} \right|< \epsilon[/math] QED.

(this is my prof from calc 1 and 2 in the picture)

Sorry I meant to write [math] N( \epsilon) := max \{ N_{1}( \epsilon),N_{2}( \epsilon) \} [/math]

>>9134112
This lmao. Brainlets btfo'd

>>9134812
Guy wants it with ε. Is the proof I provided wrong?

File: tfw too smart.png (426KB, 1440x2560px) Image search: [Google]

426KB, 1440x2560px

>>9134007
>>9134034
>>9134046
>>9134080
>>9134112

I agree with these guys, I would seriously recommend you find a different learning resource. This solution is pretty terrible, even for an introductory analysis text. My advice would be to buy Mattuck's "Introduction to Analysis." That book was miles more intuitively informative than Rudin or Abbott, so there's that.

As for what he did after that part you mentioned. Observe that:
| x - L | < e
=> -e < x - L < e, for e < 0. The rest is simple arithmetic that follows. But I wouldn't solve the problem that way to begin with. As Mattuck outlines in the first few chapters of his book, sequence and function approximation is one of the most fundamental tools of analysis, and it would serve you well to have an intuitive understanding of how to use it.

Notice that:
| (t+2)/[(t-1)(t-3) | = | t/[(t-1)(t-3) + 2/[(t-1)(t-3)] |
< | t/(t^2) + 2/[(t-1)(t-3) |, for t large

And lim { t/(t^2) + 2/[(t-1)(t-3) } = 0. I hope this helps.

>>9133582
This is the most retarded thing I've seen today.

>>9134177
Nigga you can't just split the absolute value like that, I think you meant to use inequality.

>>9134875
Fuggggggg that approximation was actually wrong in retrospect, sec lemme think of another one

>>9133582
Do you even understand what a limit is? It looks like you think mathematics is just a bunch of random symbols, and if you can string enough of them together you'll pass.

>>9134875
>>9134898
I suppose what I was really trying to say was that t < (t-1)(t-3) for t large. ¯\_(ツ)_/¯

>>9134905
spotted the retard

>>9134911
I guess the answer is no, then. You don't understand what a limit is.

>>9134911
not him, but trying to put t as a function of epsilom is pretty silly in this case

>>9134881
Yeees I meant to write < . Thanks

>>9134931
His problems starts way sooner than that. To do a proper ε-N proof you need to let ε>0 be given, and then show that |a_t| is less than this ε for all t big enough. OP, on the other hand starts by assuming that |a_t| is less than ε. Not only that, his next sentence is "Thus," and then a calculation that doesn't even follow from his wrong assumption. The fact that he then ends the whole thing by using the quadratic formula (which is also pretty fucking stupid), is barely even worth mentioning.

>>9134947
oh wow my bad
I just skipped straight to the moduli and assumed he at least started in the right direction

>>9134947
So the proof i provided is incorrect?

>>9134988
the a_t should be a_n in the formal defn

the "we therefore start by saying that |a_t| < epsilom" doesnt really make any sense at this point. You havent put any constraints on t so I can say t = 4 and epsilom = 0.000001
then |a_t| = 20/18 which is greater than 0.000001, so this line is incorrect

>>9134988
As I said, you're just stringing random symbols together. You have no idea what a limit nor what an ε-N proof is.