Question

Write the precise definition of the limit (Do not prove the limit).

Answer 1

\[\lim_{x \rightarrow3^- }\frac{ 1 }{ \sqrt{3-x} }= \infty \]

Answer 2

The definition of limit:
\[\lim_{x \rightarrow a} f(x)=b \iff Fe E(b, \epsilon) \exists E^*(a, \delta) / \forall x \in E^* (a, \delta): f(x) \in E(b, \epsilon)\]
\[\delta, \epsilon \in \mathbb{R}\]

Just make the given limit fit this definition and ou are good to go.

Answer 3

I find that definition difficult to read.
What does E and F stand for

Answer 4

there exists and for all, I would guess

Answer 5

Fe= (Not iron) "For every"

\(E^* (a, \delta)\) = "Closed enviroment with center 'a' and radius delta"

\(E(b, \epsilon)\)= "Open enviroment with center 'b' and radius epsilon"

Answer 6

The standard definition for limit, in terms of delta,epsilon is given here
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html
maybe you can show how your definition is equivalent

Answer 7

not saying yours is wrong, i am interested in alternative definitions :)

Answer 8

Ah, I see, the interval notation of limit.
The very definition of "enviroment" defines an interval whose center will be the tendency of "x" and the "b" value which is the image of "a".
So, rewriting the enviroments:
\[E^*(a, \delta)= \left| x- a \right|\]
\[E(b, \epsilon)=\left| f(x)-b \right|\]
replacing everything as those intervals would be too long and scary, so we will just write it as:
\[\lim_{x \rightarrow a}f(x)=b \iff Fe \epsilon>0 \exists \delta >0 / \forall x, 0<\left| x-a \right|< \delta , \left| f(x)-b \right|< \epsilon\]

Answer 9

what is the meaning of / in your notation

Answer 10

"/" = "such that".

Answer 11

ok i agree with your definition, but i believe you should switch the open/closed environment for delta and epsilon

Answer 12

Oh, there are many types of ways in notating limits, but I agree, the last one seems less scary haha.

Answer 13

also we did not address the original question
the limit here is 'infinity', so we need to modify the definition

Answer 14

How about we tackle it together?

Answer 15

Let's now define the "limit equal infinity", you will find that then, our aim is in the Y-axis, and we can define infinity as a number in the real plane, whose norm will always be greater or less (this last one in case of minus infinity) than an arbitrary number "A".

With this established, we can then define the case of infinite limits where the image of any "x" in the interval \(\left| x-a \right|\) will just be greater as they approach "a".
Then:
\[\lim_{x \rightarrow a} f(x)= +\infty \iff Fe A>0 \exists \delta>0/ \forall x \in \left| x-a \right| : f(x)>A\]

Answer 16

$$ \large \lim_{x \rightarrow a} f(x)= +\infty \
\\~\\ \large \iff \\~\\ \forall A>0~ \exists ~\delta>0~/ ~\forall x \in ~0 < \left| x-a \right| < \delta : f(x)>A$$

Answer 17

you need to place an upper bound on the |x - a | part.
the lower bound of any absolute value expression is zero.
but note that would occur when x =a , and in the limit definition
we don't care what actually occurs at x = a . (unlike in the continuity definition)

Answer 18

in your first definition you accounted for this

Answer 19

For that reason I prefer the enviroment notation, due to being "closed enviroment" all the numbers defined by the radius except the center.
And epsilon and delta are infinitesimal values, meaning they are very very small.

Answer 20

wouldn't that be an open environment, as you are excluding the center x=a ?

Answer 21

No no, an open enviroment includes the center.

Answer 22

ok
i guess i was thinking of set interval terminology
a closed set we include the endpoints, in an 'open' set we exclude the endpoings

Answer 23

endpoints* , lol

Answer 24

Yeah, that brings a lot of confusion haha.
So, I think we can consider the question answered?

Answer 25

yes :)

Answer 26

We make a good team.

Answer 27

indeed