what-is-limit-also-where-we-use-it-

Question Number 155331 by zakirullah last updated on 28/Sep/21

what is limit ? also where we use it .

Answered by TheHoneyCat last updated on 29/Sep/21

The idea of l i m i t is very very general .

It is present in post fields of math

(all of the non - descreet ones, but also some others)

The general idea of limits is “ Looking at

things that get closer to a given value with {time}^{″}

This is the case for a squence whose limit is a_{\infty}

\lim a_{n} = a_{\infty} is not a formula, but a statement

defined as folows :

\forall d > 0 \exists N \forall n > N (- d) < a_{n} - a_{\infty} < d

which means that for evey distance you want

to consider (\forall d > 0) at some point all the

values of the seqence (\exists N is that point

\forall n > 0 a r e t h e v a l u e s t h a t f o l o w) will be closer

to a_{\infty} than d .

this of course can be generalised for any

kind of space you can imagine (i n t o p o l o g y)

if you have a distance function d (such that

d (x, y) is the distance between x and y)

we get

\forall ε > 0 \exists N \forall n > N d (a_{n}, a_{\infty}) < ε

a l s o, i t {d o s n}^{'} t h a v e t o b e a s e q u e n c e

let f be a function from a space with a distance

d_{1} to another with a distance d_{2}

\lim_{x \to x_{0}} f (x) = y_{0}

m e a n s :

\forall ε_{2} > 0 \exists ε_{1} > 0 d_{1} (x, x_{0}) < ε_{1} \Rightarrow d_{2} (f (x), y_{0})

still the same idea, if you go close enought to x_{0}

the function will go close enought to y_{0}

That is a limit and since it is very general, it

is used almost everywere in both math and

physics

but If you want some examples here are a few

(note that usualy the distance d (x, y) i s ∣ x - y ∣)

this defines what a continious function is

indeed, a function is continuous on X when

\forall x_{0} \in X \lim_{x \to x_{0}} f (x) = f (x_{0})

this is also how you define the derivative

f^{'} (x_{0}) = \lim_{x \to x_{0}} \frac{f (x) - f (x_{0})}{x - x_{0}}

hence how you define speed, and acceleration

wich are the fundamentals of physics

You can do infinite sums witch is essential in

may calculations (it assures you that things

{don}^{'} t blow up)

I hope this answers your question

also, I skiped over this because it is technicaly

not the same thing, but in the case of functions

that have values in R (so not every function)

you can define “ \lim_{x \to + \infty}^{″}

and also “ = + \infty^{″}

it is a bit different since \infty is not a number

and you are never “ c l o s e t o {i t}^{″}

but the idea is just to switch the “ g e t t i n g c l o s e {y o}^{″}

with a “ i s n e v e r {b o u n d e d}^{″}

I live it yo you as an exercise if you want .

: -)

Commented by zakirullah last updated on 29/Sep/21

so great sir! A boundle of thanks .

Commented by TheHoneyCat last updated on 29/Sep/21

Your welcome ��