## anonymous 5 years ago What is limit?

1. nowhereman

Do you know what a infinite sequence is?

2. anonymous

( I am only 14 years odl yet, And want to learn earlier :) )

3. anonymous

hmm I don2T know its turkish .. explain ?

4. nowhereman

Basically it's a function from the the natural numbers into any other set. (e.g. real numbers). So for every number/position 1,2,3,... you have an element of your set. That's just what's meant by sequence. And infinite, because the natural numbers don't end, so neither does your sequence. For example you could have a constant sequence 7,7,7,7,... or you could have the natural numbers themselve: 1,2,3,4,... or you could have 1/ all even numbers: 1/2, 1/4, 1/6, 1/8, ... All those are infinite sequences.

5. anonymous

ok i got it

6. nowhereman

Limits come to live, as you want to know how these sequences behave at infinity. If the limit of a sequence exists, the sequence tends more and more to that limit. For example the last sequence there 1/2, 1/4, 1/6, ... goes to 0. In more rigour terms you can say, that no matter how small an ε > 0 you choose, the sequence will eventually stay closer to the limit than that ε.

7. anonymous

hmmm so how can we use these?

8. anonymous

we do the process near of the "lim" keyword till the count under of it arives to the limit?

9. nowhereman

The most noted application is with derivations. I guess you know about slope of straight lines of the plane. For general functions, it is much more difficult to give a slope and of course it's value depends on the point you choose. So you start by drawing a secant on the function, which will cut the function graph at the points (x, f(x)) and (y, f(y)). Then you can calculate it's slope $\frac{f(x)-f(y)}{x-y}$. But that only exists for $$x \neq y$$. So you have to take a sequence of x which tends to y and see where that quotient goes. That is the slope of $$f$$ at the point $$x$$.

10. anonymous

ahh ok i got it

11. nowhereman

So well, there are two types of limits, to be fair. The one I showed you first was for a sequence, if you name your sequence $$a_1, a_2, a_3, \dots$$ then you can write it as $\lim_{n→∞}\;a_n$ But then you can introduce another limit, for a function $$f$$ (so maybe just a simple term) and write $\lim_{x→x_0}\;f(x) = y$ which means that for any sequence that tends to $$x_0$$ the function value tend to y:$\forall (x_n)_{n=1}^∞: \left( \lim_{n→∞}\;x_n = x_0 \right)⇒ \left(\lim_{n→∞}\;f(x_n) = y\right)$

12. anonymous

kk^^ function f =??

13. nowhereman

f can be any function

14. anonymous

ahh k ^^

15. anonymous

so it does the process near of the "lim" till the squence comes to its limit (sorry for a bad english :( )

16. nowhereman

Yes, but there is no guarantee, that the process will succeed to lead to one particular number. For the last part you also would like to have, that, as the x approaches $$x_0$$ the function values approach $$f(x_0)$$ but that must neither be the case. If it is, you say, f is continuous at $$x_0$$

17. anonymous

hmmm k^^

18. nowhereman

I guess you can find a lot on it also on wikipedia ;-)

19. anonymous

:-) ty much ^^

20. anonymous

and can you help me in my other question?

21. nowhereman

Let's see.