SparklyUnicorn Posted January 19, 2017 Share Posted January 19, 2017 (edited) 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 Quote Link to comment Share on other sites More sharing options...
chiguirre Posted January 19, 2017 Share Posted January 19, 2017 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. 1 Quote Link to comment Share on other sites More sharing options...
kiana Posted January 19, 2017 Share Posted January 19, 2017 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. Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 19, 2017 Author Share Posted January 19, 2017 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. Quote Link to comment Share on other sites More sharing options...
regentrude Posted January 19, 2017 Share Posted January 19, 2017 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. 3 Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 19, 2017 Author Share Posted January 19, 2017 Although that explanation does support what he said. Quote Link to comment Share on other sites More sharing options...
kiana Posted January 19, 2017 Share Posted January 19, 2017 (edited) 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 1 Quote Link to comment Share on other sites More sharing options...
regentrude Posted January 19, 2017 Share Posted January 19, 2017 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. 1 Quote Link to comment Share on other sites More sharing options...
regentrude Posted January 19, 2017 Share Posted January 19, 2017 (edited) 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 3 Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 19, 2017 Author Share Posted January 19, 2017 so what in heck is my son confused about he feels pretty adamant about this Quote Link to comment Share on other sites More sharing options...
kiana Posted January 19, 2017 Share Posted January 19, 2017 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"? 1 Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 19, 2017 Author Share Posted January 19, 2017 I'll look at his notes. Let me see if I can post a picture of a problem. Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 19, 2017 Author Share Posted January 19, 2017 Ok...I found his confusion in his notes. Thank you! 1 Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 20, 2017 Share Posted January 20, 2017 (edited) 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 Quote Link to comment Share on other sites More sharing options...
kiana Posted January 20, 2017 Share Posted January 20, 2017 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) Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 20, 2017 Share Posted January 20, 2017 Yeah, and limx->0 (sin (Ï€/x) ) is another great example where the limit does not exist, in case anyone is cluing on fractions... :001_smile: 1 Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 20, 2017 Author Share Posted January 20, 2017 added another question.... Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 20, 2017 Author Share Posted January 20, 2017 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. Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 20, 2017 Share Posted January 20, 2017 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 limx->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... 1 Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 20, 2017 Author Share Posted January 20, 2017 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 limx->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. Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 20, 2017 Share Posted January 20, 2017 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... 1 Quote Link to comment Share on other sites More sharing options...
SparklyUnicorn Posted January 20, 2017 Author Share Posted January 20, 2017 Yeah reading whatever I can find on it....I don't know what led him to believe that. Really, no clue. He's stubborn. Quote Link to comment Share on other sites More sharing options...
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