Toby Opferman
http://www.opferman.net
toby@opferman.net


   				  Limits

Welcome to the tutoiral on limits. This will try to provide a basis for learning
what a limit is.



A limit is what a function approaches as it approaches some number.
Here is an example:


limit f(x)
 x->Infinite


That is "The limit of f(x) as x approaches Infinite"

If f(x) = 1/x  then, 

1/2 = .5
1/3 = .33
1/4 = .25 
1/100 = .01
....

Notice how the numbers get smaller. Now plug in infinite to x and imagine what
number the function f(x) approaches.

limit f(x)       = 0
 x->Infinite 

The limit is 0. No, the function will never be zero, but it doesn't have to be.
It never converges to 1 number, it keeps getting closer and closer to the x axis,
closer to 0.  Therefore, the limit of the function is 0.


Look at this function now:

limit x^2 
  x->Infinite


This function has no limit.  The limit is infinite.

One of the best ways to do a limit is plug in the number.  If the number
does not exist at that point, then look at numbers before it and after it.

If numbers before it go towards a different limit than the ones that go towards it 
after the number, it is said to have a Left and a Right limit.


Let us take the old function again f(x) = 1/x


limit 1/x
  x->0

Plug in 0, and you get undefined.

Now, plug in numbers between 1 and 0

1/1 = 1
1/.5 = 2
1/0.000000000001 = 1000000000000


Looks like Infinite from the right, so there is no right limit.
Let us try the left.


1/-1 = -1
1/-.5 = -2
1/-0.000000000001 = -1000000000000

Looks like -Infinite from the left, so there is no left limit.



Remeber, it must converge to a certain number or BE that certain number
for that to be the limit.


limit 1/x    = 1
  x-> 1

The limit to 1/x while x approaches 1 is 1.   1/1 = 1.

The limit is either the number it approaches or the number it lands on.

Some examples:

limit 2/3x    = 1/3
  x->2



limit |x - 3|/(x - 3)
 x->3
  

The limit of the above, you get 0/0 = undefined.
If you approach the function from the left:
|2 - 3|/(2 - 3) = 1/-1 = -1
|1 - 3|/(2 - 3) = 2/-2 = -1

You see that, From the left it approaches -1.

But, if you approach the function from the Right:
|4 - 3|/(4 - 3) = 1/1 = 1
|5 - 3|/(5 - 3) = 2/2 = 1

Thus, from the right it approaches 1.

We say that the "limit does not exist"  But, if we were just looking
for a 1 sided limit we could say:

Right Limit
lim f(x) = 1
 x->3 +

Left Limit
lim f(x) = -1
 x->3 -

*Notation*
The + means approach from the right and the - means approach from the left.


It is pretty simple really.  It's just finding where a function is continous,
and what value does it have at the value it is approaching, or what is the value
it is tring to approach.