Ytukyg

I am trying to formally understand the sequential definition of a limit.

My focus, in particular, is when $x rightarrow ∞$ and when $limlimits_{x to ∞}{f(x)}=∞$.

"$delta - epsilon$" explanations will work as well, though I am trying to focus on the sequential aspect of it.

I borrowed the following definition from here.

Let $a∈R$, let $I$ be an open interval which contains $a$, and let $f$ be a real function defined everywhere on $I$ except possibly at $a$. Then $limlimits_{x to a}{f(x)}=L$
exists if and only if $f(x_n)→L$ as $n→∞$ for every sequence $x_n∈I$∖{$a$} which converges to $a$ as $n→∞$.

Is there a way to say this without using the "$∞$" symbol?

I interpret "$x rightarrow a$" to mean: "Every sequence, $x_n$, in $I-${$a$} that converges to $a$, as $n$ gets arbitrarily large."

The only thing I can come up with to say is: "There exists some $N$ in the natural numbers such that: For every $x_n, n>N$, $x_n$ converges to $a$." But i'm not sure if that's correct.

Also, I interpret "$x rightarrow ∞$" to mean: "Every sequence, $x_n$, in $I$ that diverges".

Next, what, exactly, does it mean when $L=∞, -∞,$ and $+∞$ respectively?

My interpretation is just that $f(x_n)$ diverges. Again, how would I say that more formally?

Any insights here would be greatly appreciated.

A sequence of real numbers ${x_n}_{n=1}^infty$ converges if there is $a in mathbb{R}$ satisfying: for every $varepsilon>0$ there exists a positive integer $N$ such that $|x_n-a|<varepsilon$ whenever $n geq N$. In the case defined above we say that ${x_n}_{n=1}^infty$ converges to $a$ and we may write $x_n to a$ as $n to infty$.

It is important to notice that the positive integer $N$ mentioned in the definition above will depend both on $varepsilon>0$ and the sequence ${x_n}_{n=1}^infty$. In example; The sequences $big{frac{1}{n}big}_{n=1}^infty$ and $big{-frac{1}{3n}big}_{n=1}^infty$ both converge to $0$. For each sequence, can you find positive integers $N_1$ and $N_2$ such that the definition of convergence is satisfied given $varepsilon_1=frac{1}{10}$ and $varepsilon_2=frac{1}{100}$?

A function $f$ becomes infinite as $x to a$ if for every $M>0$ there exists $delta>0$ such that $|f(x)|>M$ whenever $0<|x-a|<delta$. For this limit we may write $f(x) to infty$ as $x to a$.

Just a quick formal focus on one of the OP's questions:

Definition: A sequence $(x_n)_{n in mathbb N}$ of numbers in $mathbb R$ is said to approach $+∞$ if for every $B in mathbb R$ the set ${m in mathbb N ; | ; x_m le B }$ is finite.

We also say that the limit of such a sequence is $+∞$ and write $limlimits_{n to +∞} x_n = +∞$ to express this.

Definition: Let $I$ be an open interval of the form $(s,+∞) subset mathbb R$ and let $f$ be a real function defined everywhere on $I$. Then we say that $f$ approaches $+∞$ as $x$ approaches $+∞$, writing

$tag 1 limlimits_{x to ∞}{f(x)}= +∞$

to mean that for every sequence $(x_n)_{n in mathbb N}$ that approaches $+∞$, if the terms $x_n$ all belong to the set $(s,+∞)$, then the corresponding sequence $(f(x_n))_{n in mathbb N}$ also approaches $+∞$.

Proposition: The statement $limlimits_{x to ∞}{f(x)}= +∞$ is equivalent to saying that for every $B in mathbb R$ there exist $k in mathbb R$ with $(k,+∞)$ in the domain of $f$ and the image $f[,(k,+∞),]$ contained in the interval $(B,+∞)$.

Calculus students certainly know, intuitively, how to graph many functions 'out at $+∞$', but these are some formal definitions.

$a_n$ -> a as n -> oo is defined as for all open U.

if a in U, then $(a_n)$ is eventually in U.

$(a_n)$ is eventually in U when there is some

integer n with for all j > n, $a_j$ is in U.