Easy as limits question so you should help all the first year unsw actl kids (1 Viewer)

[mreditor16]

Yes to "so is M = 3 or M = 3 + 1/epsilon ? :O"

LOL so I can write either in my answer?

 

[InteGrand]

LOL so I can write either in my answer?

No.

Sorry, that "yes" was a joke referring to "or" in logic. ( http://en.wikipedia.org/wiki/OR_gate )

Use the

 

[InteGrand]

(For example, the answer to "is an apple red or purple" is "yes".)

 

[InteGrand]

(For example, the answer to "is an apple red or purple" is "yes".)

(Assuming the apple isn't a green one, but is a red one.)

 

[InteGrand]

The thing needs to guarantee that for ALL x greater than M, given ANY , we'll have .

In this case, if we just had M equals the constant 3, we could pick to be small enough so that not every x > 3 makes the function close enough to the desired limit.

(e.g. if x = 3.1 > 3, , thus failing if we pick , say)

This is why we usually need M to depend on .

 

[mreditor16]

Then, on a similar note, is this proof correct?

 

[mreditor16]

Also, I have a theory - maybe you only use the idea of min for epsilon-delta proofs but not for epsilon proofs, such as the original Q? Anyone?

 

[InteGrand]

Also, I have a theory - maybe you only use the idea of min for epsilon-delta proofs but not for epsilon proofs, such as the original Q? Anyone?

This proof was an epsilon-delta proof. Maybe give an example of where you've seen a min being used? Usually for these ones, where they're just asking you to prove a limit of a simple function, you don't need to use a min.

Edit: realised the picture had an example

 

[mreditor16]

This proof was an epsilon-delta proof. Maybe give an example of where you've seen a min being used? Usually for these ones, where they're just asking you to prove a limit of a simple function, you don't need to use a min.

look at post 26, which is how my lecturer did it.

 

[photastic]

Then, on a similar note, is this proof correct?

I don't get where the 1/5 popped out

 

[InteGrand]

I don't get where the 1/5 popped out

-1/5 is the value of the quotient when x = -1.

 

[mreditor16]

I don't get where the 1/5 popped out

for that range? kress explained it in lecture!

 

[InteGrand]

Then, on a similar note, is this proof correct?

OK I think I get what the lecturer's saying now:

If , taking always makes the function within units of the limit.

If , taking does not always guarantee the function will be within units of the limit, because our fact that we used in the proof is not valid for all x in this domain. But then if we take , it is certainly true that , so that means (since we've taken , so we can use that fact the lecturer used), and then the rest of the lines of the proof follow.

 

[InteGrand]

(So that's why taking the min guarantees the function is close enough to the limit)

 

[mreditor16]

OK I think I get what the lecturer's saying now:

If , taking always makes the function within units of the limit.

If , taking does not always guarantee the function will be within units of the limit, because our fact that we used in the proof is not valid for all x in this domain. But then if we take , it is certainly true that , so that means (since we've taken , so we can use that fact the lecturer used), and then the rest of the lines of the proof follow.

exactly so how does this min stuff relate to the original proof for our original Q?

 

[InteGrand]

exactly so how does this min stuff relate to the original proof for our original Q?

For the original Q, we don't need to worry about min, because any facts that you used were true for all x in the domain you restricted x to (plus, that was a limit to infinity, so you can just restrict x to be greater than 3, or greater than anything, without having to worry about the restriction).

 

[mreditor16]

For the original Q, we don't need to worry about min, because any facts that you used were true for all x in the domain you restricted x to (plus, that was a limit to infinity, so you can just restrict x to be greater than 3, or greater than anything, without having to worry about the restriction).

so essentially only worry about min for questions proving limit to a certain point, whereas don't worry about for questions proving limit to infinity ???

 

[InteGrand]

so essentially only worry about min for questions proving limit to a certain point, whereas don't worry about for questions proving limit to infinity ???

Not necessarily. For easier limits at a point, you may not need to worry about min at all. For trickier ones, you usually need to use more tricks to prove the limit, and one of these tricks is to use "min". It all depends on the question specifically, really.

 

[mreditor16]

Thanks so much guys!!! especially you Integrand