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You are not logged in. #1 20060123 06:34:28
limitsFind the limit of sqrt(x^2)/x as x approaches zero. I figured this is the same as x/x which is always 1, but according to the answer, the limit is undefined. Now sqrt(x^2) as is, is the same as x, and x/x does not have a limit as x approaches zero, since the right hand limit doesn not equal the left hand limit. A logarithm is just a misspelled algorithm. #2 20060123 06:41:21
Re: limitsYou can always simplify, but in this case, you just simplified wrong. "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..." #4 20060123 08:12:02
Re: limitsRicky, isn't the problem really that there are so many instances in which we use the identity √x² = x and it winds up working for our purposes anyway. Technically, doesn't √x^2 = x andx? I think that this is where most people just get mentally complacent. #5 20060123 08:47:49
Re: limits
If √x² = x works, then so will √x² = x. Only one of them is correct however...
I think you meant the piecewise function: "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..." #8 20060123 09:27:12
Re: limitsSure is. Like I said in that other post, x^2 isn't a 11 function. Since it's not 11, you can't figure out what x was from y. "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..." #9 20060123 09:56:34
Re: limitsby convention √x means the positive square root of x. Not just the square root of x. The negative square root is simply represented by √x. Last edited by mikau (20060123 10:01:10) A logarithm is just a misspelled algorithm. #10 20060123 14:20:55
Re: limitsI got this from Wikipedia: #11 20060123 14:28:53
Re: limitsI believe the standard is that the √x is always positive. If you want both roots, you simply put ±√x. If you take the root in the middle of a problem, you must put the ± in: Last edited by Ricky (20060123 14:29:28) "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..." #12 20060123 14:34:08
Re: limitsRicky, I respect your input. I'll take what you said at face value. I have never heard that before. From now on I will only use the positive root unless otherwise stated. #13 20060123 14:37:32
Re: limitsirspow, there is nothing wrong with arguing. In fact, that's normally how progress is made. If you disagree, then please, disagree with me. I can be just as wrong as anyone. The way I was taught is to take only the positive unless otherwise stated. In my experience, it has worked out. If you can find an example where it doesn't (an answer is missed), then post it here. Last edited by Ricky (20060123 14:40:41) "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..." #14 20060123 14:44:26
Re: limitsWell, thanks for the discussion on this point, it was informative. I will do more research on this topic to see if there truly is some consensus. Maybe I just forgot something that I learned a long time ago. I am 34 years old and was introduced to such concepts a long time ago. If I come up with something new I will post it in a more appropriate area of the forum. If not, then I will atleast learn from my research. #15 20060123 15:31:14
Re: limitsricky is right. Thats the same thing I read. Like I said, by convention, √x means the positive square root. Where as if you were to verbally ask for the square root, it could mean the positive or negative root. A logarithm is just a misspelled algorithm. #16 20060124 08:09:37
Re: limitsI said I would post somewhere else if I came up with something new, but what I am going to say here doesn't seem to justify another thread. The most reasonable explanation that I have come across, and it is completely logical now that I have heard it in just this way, is this: 