Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2006-01-23 06:34:28

mikau
Super Member

Offline

### limits

Find 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.

This worries me, how do you know when you can simplify and rearrange a limit, and when you can't?

A logarithm is just a misspelled algorithm.

## #2 2006-01-23 06:41:21

Ricky
Moderator

Offline

### Re: limits

You can always simplify, but in this case, you just simplified wrong.

√x² ≠ x, this is never valid, whether dealing with limits or not.

√x² = |x|

"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..."

## #3 2006-01-23 06:51:47

mikau
Super Member

Offline

### Re: limits

A logarithm is just a misspelled algorithm.

## #4 2006-01-23 08:12:02

irspow
Power Member

Offline

### Re: limits

Ricky, 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 and-x?  I think that this is where most people just get mentally complacent.

## #5 2006-01-23 08:47:49

Ricky
Moderator

Offline

### Re: limits

Ricky, 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.

If √x² = x works, then so will √x² = |x|.  Only one of them is correct however...

Technically, doesn't √x^2 = x and-x?

I think you meant the piece-wise function:

√x^2 = x if x > 0
√x^2 = -x if x ≤ 0

In which case, you'd be right.

"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..."

## #6 2006-01-23 08:58:56

irspow
Power Member

Offline

### Re: limits

What I meant specifically above is that √16 is generally accepted to be 4, but in reality it can not be determined whether the original root was 4 or -4.  Meaning 4 and -4 are both possible roots of √16.

Last edited by irspow (2006-01-23 08:59:34)

## #7 2006-01-23 09:01:56

irspow
Power Member

Offline

### Re: limits

That would be like determining if the 16 were from -4² or 4².  I don't see how this would be possible.

Isn't this the reason that the quadratic equation places the ± before the discriminant?

Last edited by irspow (2006-01-23 09:05:52)

## #8 2006-01-23 09:27:12

Ricky
Moderator

Offline

### Re: limits

Sure is.  Like I said in that other post, x^2 isn't a 1-1 function.  Since it's not 1-1, 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 2006-01-23 09:56:34

mikau
Super Member

Offline

### Re: limits

by 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.

We say √2^2 is 2, since by convention, the expression asked for the positive square root. If you ask for √(x^2), we want the positive square root, which will only be positive x. Not negative x. Its permissable to move a squared positive constant to the outside of the radical, but a squared variable could either have been a positive or negative number. Either way, its square would be positive, and the square root of a positive number is always positive, thus  √x^2 = |x|

Last edited by mikau (2006-01-23 10:01:10)

A logarithm is just a misspelled algorithm.

## #10 2006-01-23 14:20:55

irspow
Power Member

Offline

### Re: limits

I got this from Wikipedia:

There are two solutions to the square root of a non-zero number. For a positive real number, the two square roots are the principle square root and the negative square root. For negative real numbers, the concept of imaginary and complex numbers has been developed to provide a mathematical framework to deal with the results.

However, I will allow that the [b]PRINCIPAL[b] √x² = |x|.  I am just wondering

how one  is to determine which root is needed for a given problem when it is not stated

explicitly in the problem.  Do we always assume that a problem wants the principle root unless

it is specified that both roots are needed?

## #11 2006-01-23 14:28:53

Ricky
Moderator

Offline

### Re: limits

I 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:

x² = y
±√x² = ±√y
±|x| = ±√y
x = ±√y

Last edited by Ricky (2006-01-23 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 2006-01-23 14:34:08

irspow
Power Member

Offline

### Re: limits

Ricky, 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.

I was not trying to argue with you on this point, I really wasn't sure why the other root was being ignored.  Thanks for your insight.

## #13 2006-01-23 14:37:32

Ricky
Moderator

Offline

### Re: limits

irspow, 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.

But seriously, never take what anyone says at face value unless you agree with it.

Last edited by Ricky (2006-01-23 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 2006-01-23 14:44:26

irspow
Power Member

Offline

### Re: limits

Well, 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.

Thanks again.  Oh, I never accept what others say as fact, just another observation,  but convention of society says to be polite.

## #15 2006-01-23 15:31:14

mikau
Super Member

Offline

### Re: limits

ricky 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.

But of course, if you have the expression x^2 = 4, you can take the square root of both sides but you must consider the postive and negative square root. x = +- √4     note! If √4 represented the positive AND negative square root of 4, then the +- signs would be uneccessary and redundant.

A logarithm is just a misspelled algorithm.

## #16 2006-01-24 08:09:37

irspow
Power Member

Offline

### Re: limits

I 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:

We use the principle root whenever the radical is within a function, because as most of us know, a function can have only one output value for every one input value.  So whenever you are taking derivatives or integrals or merely changing the independent value within a function, the root used will always be the principle (or positive) root.

Whenever we are not dealing with a function relating a independent variable to a dependent one, then both the principle and negative roots must be considered.

Now everything is crystal clear to me.  These are the reasons why we have that "convention".  I just didn't make the connection before looking into it.

I hope that this clears this up for everyone like me who might not have grasped exactly why we have the "convention" that we do.