Why is Zero the Limit?

Date: 02/18/2002 at 14:23:02
From: Kelly Driskill
Subject: Infinite Sequences and Series

My question is, why is zero called the limit of the terms in the 
sequence the limit of 1 over n, as n approaches infinity, equals zero?

Date: 02/18/2002 at 21:13:43
From: Doctor Roy
Subject: Re: Infinite Sequences and Series

Hello,

Thanks for writing to Dr. Math.

The limit is zero, because each successive term is closer to zero than 
the previous one.  Consider 1/n as n get bigger.

1/10 = 0.1
1/100 = 0.01
1/1000 = 0.001

You can see that 1/n gets smaller and smaller as n gets bigger.  So, 
the limit, as n becomes infinitely large is 0.  Of course, since n can 
never reach infinity (infinity is NOT a number), 1/n never quite makes 
it to zero. However, it gets arbitrarily close to 0, which is good 
enough for a limit.

I hope this helps.

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/25/2002 at 22:08:00
From: Katie Meyer
Subject: Limitations

1) Zero is called the limit of the terms in a sequence. Why? 
   For example: lim 1/n=0
                n -> infinity

2) When the limitation of the problem lim 4n+2
                                      n -> infinity
   is 4, what does that mean?

Date: 02/26/2002 at 11:19:36
From: Doctor Ian
Subject: Re: Limitations

Hi Katie,

In each case, we can plot some values of n, and f(n).  For comparison, 
let's include a couple of other functions, too:

  n     1/n      (4n+2)/n     3n + 6    sin(n)
 ----   ------   ---------    ------    ------
  1     1        6              9       0.841
  2     1/2      5             12       0.909
  3     1/3      4 2/3         15       0.141
  4     1/4      4 1/2         18      -0.757
  5     1/5      4 2/5         21      -0.959
  6     1/6      4 1/3         24      -0.279

Now, as we keep increasing the value of n, the third function just 
keeps increasing. And the fourth function will flop around between 
-1 and 1. You should try more values of n (including some large ones, 
like n = 100, n = 1000, n = 1,000,000, and so on) to convince yourself 
that this is true. 

But each of the first two functions seems to get closer and closer to 
a 'final value'. The first function gets closer and closer to 0, 
although it never quite gets there. And the second function gets 
closer and closer to 4, although it never quite gets there, either.

But we can imagine that, if we could keep increasing n forever, the 
functions _would_ reach 0 and 4 respectively. And this is what we mean 
when we say something like 

  the limit of f(n) = (4n+2)/n, as n approaches infinity, is 4

That's too much to write over and over, so we abbreviate it

  limit   (4n+2)/n = 4
   n->inf
    
Normally, instead of 'inf', we would use the symbol for infinity, 
which looks like an '8' lying on its side. 

Another way to think of it is that we can make (4n+2)/n get as close 
as we want to 4, by making n as large as we need to... but no matter 
what we do, we'll never get all the way to 4. Which makes 4 the 
'limit' to how low we can go. 

In a rough sense, this is what it means for a function to have a 
limit. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/