what is limit? also where we use it.


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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 ��
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