Sunday, September 22, 2019 | Toby Opferman


Toby Opferman


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)

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

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 

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

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

limit |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 -

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.

About Toby Opferman

Professional software engineer with over 15 years...

Learn more »
Codeproject Articles

Programming related articles...

Articles »

Resume »

Email: codeproject(at)opferman(dot)com