question about Limits (math) ADDED to POST #1

[SparklyUnicorn]

Limits cannot be infinity right? 

 

Might be a super stupid question, but I copied down several examples where the teacher wrote infinity as the answer.  DS came home from his class and said UHHH no.  That can't be.  So indeed..no I don't think so.

 

That's rather disconcerting....

 

-----

 

Ok, so is there some sort of philosophical debate over whether or not infinity is a possible "limit" on a limit?  Because my kid is pretty adamant and did show me a few things that suggest that.  He is adamant his instructor told them that.

 

Use very small words when talking to me about this.  LOL  I'm learning this stuff too and really don't know. 

Edited January 20, 2017 by SparklyUnicorn

[chiguirre]

Yes, they can. Here's an explanation:

 

http://tutorial.math.lamar.edu/Classes/CalcI/InfiniteLimits.aspx

 

This is one of the few things that stuck in my brain from Calc class. If you have the limit of a fraction with x in the denominator, the answer is infinity.

[kiana]

Um. 

 

So what we mean when we say that a limit is infinity, we mean that it grows larger than any possible bound. 

 

Some people informally and imprecisely will write infinity when a limit grows larger in absolute value than any possible bound (for example, 1/x as x approaches 0) but this is not correct. 

[SparklyUnicorn]

Yes, they can. Here's an explanation:

 

http://tutorial.math.lamar.edu/Classes/CalcI/InfiniteLimits.aspx

 

This is one of the few things that stuck in my brain from Calc class. If you have the limit of a fraction with x in the denominator, the answer is infinity.

 

Ok.  But none of them were limits of fractions. 

[regentrude]

Yes, there can be a limit of infinity.

 

You can define that lim f(x), for x approaching a = infinity if for all N, there exists a delta such that 0<|x-a|< delta   f(x)>N.

f(x) can become arbitrarily close to infinity as x gets closer to a.

Likewise, you can define a limit of negative infinity.

[SparklyUnicorn]

Although that explanation does support what he said. 

[kiana]

This is one of the few things that stuck in my brain from Calc class. If you have the limit of a fraction with x in the denominator, the answer is infinity.

 

Careful. Not quite. For example, 7x/x tends to 7 as x tends to 0. (sin x)/x tends to 1 as x tends to 0. 

Edited January 19, 2017 by kiana

[regentrude]

Ok.  But none of them were limits of fractions. 

 

??? You can have limits of fractions that are infinity.

First example I have in my math text is f(x)=1/(x-2)^2 for x approaching 2.

[regentrude]

Now, if there is stuff like 1/x, that would only have a one-sided limit as x approaches zero, because it makes a difference whether you come from the positive or the negative side - so the actual limit does not exist.

 

But the limit of 1/x^2 for x to zero does exist and is infinity.

Edited January 19, 2017 by regentrude

[SparklyUnicorn]

so what in heck is my son confused about

 

he feels pretty adamant about this

 

 

[kiana]

so what in heck is my son confused about

 

he feels pretty adamant about this

 

I wonder if he's mixed up "don't write 1/0 = infinity" with "don't write a limit that turns into 1/0 = infinity"?

[SparklyUnicorn]

I'll look at his notes.

 

Let me see if I can post a picture of a problem.

 

 

[SparklyUnicorn]

Ok...I found his confusion in his notes.

 

Thank you!

[Mike in SA]

Um. 

 

So what we mean when we say that a limit is infinity, we mean that it grows larger than any possible bound. 

 

Some people informally and imprecisely will write infinity when a limit grows larger in absolute value than any possible bound (for example, 1/x as x approaches 0) but this is not correct. 

 

I think this is a bit confusing as written.  You might need to elaborate a bit for the sake of the readers.

 

I believe the point that you are trying to make here is that it is possible to NOT have a limit, just as it is possible to have a limit at a point where a function is undefined...

Edited January 20, 2017 by Mike in SA

[kiana]

I think this is a bit confusing as written.  You might need to elaborate a bit for the sake of the readers.

 

I believe the point that you are trying to make here is that it is possible to NOT have a limit, just as it is possible to have a limit at a point where a function is undefined...

 

Actually it was that I've seen people (sometimes people who should have known better -- like grad students) write that 1/x -> infinity as x -> 0 which is not correct. I thought that maybe that was what the teacher had done that her son had an issue with.

 

Although I probably should've clarified why it was actually wrong. :)

 

(For any curious observers, it's example 1 in the link chiguirre shared, and typeset there far better than I could do here)

[Mike in SA]

Yeah, and lim_x->0 (sin (Ï€/x) ) is another great example where the limit does not exist, in case anyone is cluing on fractions... :001_smile:

[SparklyUnicorn]

added another question....

[SparklyUnicorn]

At least I was able to convince him that it is often noted as negative or positive infinity.  Ya know, in other words, that's a thing.  But he did show me some stuff that claims it's not a thing in the strictest sense.  Which, I would not know.  So I said just make sure to ask your teacher again about that and how she wants you to handle it.

 

I don't mind debating stuff with him, but man can he get intense. 

[Mike in SA]

Ok, so is there some sort of philosophical debate over whether or not infinity is a possible "limit" on a limit?  Because my kid is pretty adamant and did show me a few things that suggest that.  He is adamant his instructor told them that.

 

Use very small words when talking to me about this.  LOL  I'm learning this stuff too and really don't know. 

 

Yes, infinity is a possible limit.

People often try to think of infinity as an element of the real numbers.  It is not.  Infinity is a concept, not a number.  It means "greater than any chosen number." 

 

So, let's test his premise, based on the definition - "greater than any chosen number":

 

We'll play a little game with lim_x->0(abs(1/x)).

You pick a number and a range, and then I get to pick a smaller number. 

You just find a number "a" and a range {y: r₁ < y < r₂} such that abs(1/a) is contained within the range.

 

My objective is to find a number "b" such that abs(1/b) > r₂

If I find it, then I win.

 

 

So, for infinity to NOT be the limit, you must be able to find a number "a" such that I cannot win. 

 

(^The above describes the definition of a limit. One of the reasons I am disappointed in calc AB/BC is that they fail to teach this properly.)

 

 

 

Here's the catch, I will tell you - right now - that I hereby choose b = 1 / 2r₂.  I'm just going to wait my turn, after you pick your range.

 

abs( 1 / ( 1 / 2r₂ ) ) = abs( 2r₂) = 2abs(r₂)

Since 2abs(r₂) > abs(r₂), I win, no matter what you pick.

 

Therefore, the limit exceeds every finite element of the real numbers.  It is in-finite.  That is precisely the definition of infinity.

 

(note: "finite" means assignable or countable.  You assigned a range of numbers, and I found one which was greater.)

 

 

 

That all may take some chewing on, but that's how limits go...

[SparklyUnicorn]

Yes, infinity is a possible limit.

People often try to think of infinity as an element of the real numbers.  It is not.  Infinity is a concept, not a number.  It means "greater than any chosen number."

 

So, let's test his premise, based on the definition - "greater than any chosen number":

 

We'll play a little game with lim_x->0(abs(1/x)).

You pick a number and a range, and then I get to pick a smaller number. 

You just find a number "a" and a range {y: r₁ < y < r₂} such that abs(1/a) is contained within the range.

 

My objective is to find a number "b" such that abs(1/b) > r₂

If I find it, then I win.

 

 

So, for infinity to NOT be the limit, you must be able to find a number "a" such that I cannot win. 

 

(^The above describes the definition of a limit. One of the reasons I am disappointed in calc AB/BC is that they fail to teach this properly.)

 

 

 

Here's the catch, I will tell you - right now - that I hereby choose b = 1 / 2r₂.  I'm just going to wait my turn, after you pick your range.

 

abs( 1 / ( 1 / 2r₂ ) ) = abs( 2r₂) = 2abs(r₂)

Since 2abs(r₂) > abs(r₂), I win, no matter what you pick.

 

Therefore, the limit exceeds every finite element of the real numbers.  It is in-finite.  That is precisely the definition of infinity.

 

(note: "finite" means assignable or countable.  You assigned a range of numbers, and I found one which was greater.)

 

 

 

That all may take some chewing on, but that's how limits go...

 

Thank you.  That is very helpful! 

 

I'll print this out and show him.

 

 

 

[Mike in SA]

If it helps, the actual definition of a limit is online at http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

 

It requires a bit of rigor, but it's actually a bit of a "squishy" thing, once you understand it.

 

By the way, three things every good calculus student should know by heart: the definition of a limit, the fundamental theorem of calculus, and the mean value theorem...

[SparklyUnicorn]

Yeah reading whatever I can find on it....I don't know what led him to believe that.  Really, no clue.

 

He's stubborn.