分类：
2010-08-31 11:08:15
Toby Opferman
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.
