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yuriythebest

Member

Registered: 2008-09-07

Posts: 3

grasping lim continuity definitions

right this is really basic but I'm still having trouble grasping it.

ok I get rule #1 - in point A the value of f(x) should not be undefined or equal positive or negative infinity.

#2- not really sure about this one- does it mean f(x) should not be "flatlined" at point A?

#3 - ????

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: grasping lim continuity definitions

1 and 2 are really just so that stating #3 makes sense.  So if you get #3, you should understand them all.

It sounds like you don't know what a limit is.  There are many examples one must see before limits really make sense, and I won't be able to present them all here obviously.  But a limit exists if, as you approach a value from both sides, you get arbitrarily close to a single point.  For example, consider the function:

We can see that:

However, because of the way f(x) is defined, the point x = 1 is undefined.  At x = 1, we get a 0 in the denominator, and we can't divide by 0.  So we have a graph that looks exactly like x-1, except the point (1, 2) is missing from the graph.  But, as x approaches 1, we see that f(x) approaches 2.  The closer and closer our x gets to 1, the closer and closer our function f(x) gets to 2.  We say that the limit as x approaches 1 of f(x) is 2.

Then of course there are right hand and left hand limits, we can go over them if you wish.  The bottom line is that for the limit at a point to exist, it must exist from both the left hand and right hand side, and they must be equal.

So #3 is saying that not only does this limit and f(a) have to exist, but it has to be that the function approaches it's value at a.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: grasping lim continuity definitions

Ricky beat me. At least I've got the definition here.

The technical definition of a limit goes like this:

In other words, you should be able to get f(x) as close to L as you like by making x get close to a.

3) just means that L from above has to be equal to f(a).

An example of a function that doesn't do that would be:

f(x) = { x if x≠ 0
         { 1 if x = 0.

There, the limit as x approaches 0 is 0, but f(0) = 1.

Last edited by mathsyperson (2008-09-10 09:03:45)

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: grasping lim continuity definitions

mathsyperson, students struggle a great deal with epsilon-delta definition after working with the "conceptual" definition of a limit for quite some time.  They won't be able to make heads or tails of the epsilon-delta definition if they don't know the conceptual definition first.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: grasping lim continuity definitions

Two things. Firstly, you actually want

here, not just

. In other words, x gets close to a but is not equal to a itself.

And secondly, you mean

.

Last edited by JaneFairfax (2008-09-10 07:57:33)