Left Limit of |x| at x=0: Conceptual Reason

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In summary, the text on page 115 of the textbook, Calculus Concepts and Contexts, says that the limit of the function -lim_{x\rightarrow 0^-}\frac {|x|}{x} does not exist.
  • #1
ktpr2
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0
This pretty book here says that [tex] \lim_{x\rightarrow 0^-}\frac {|x|}{x} [/tex] is equal to [tex] \lim_{x\rightarrow 0^-}\frac {-x}{x} [/tex] ...

I understand that we're taking a left sided limit, as x approaches 0, so x will always be negative, but we're putting that value in the absolute value function which should always return a postive result. So what's the conceptual reason for the absolute value function returning a negative number? The only thing I could come up with is the order of operation application is different; you take the absolute value of x first and then let that x approach 0 from the left, which would be result in a negative x.
 
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  • #2
It doesn't.It returns a positive value ALWAYS...Even for complex #...

Daniel.
 
  • #3
uhhhhh... then can you prove that [tex] \lim_{x\rightarrow 0}\frac {|x|}{x} [/tex] exists? :) My text, second edition Calculus concepts an contexts by james stewart, says it doesn't exist.
 
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  • #4
You mean the left or the right limit...?The partial limits are - and + infinity respectively...So the limit -------->0 (full) DOES NOT EXIST...Neither in Rbar.

Daniel.
 
  • #5
wiat a second, x can be negative; its the y value, the range, that is always positive. But even still it appears that they're abusing the || function because when you slap a function on a number you use the result, the range, not the input (otherwise there's no point in using the functino in the first place). For any of you that have this text, its on page 115.
 
  • #6
No,the limit is generally COMPUTED OVER THE DOMAIN OF DEFINITION.

Daniel.
 
  • #7
says here the partial limits are +/- 1.

If it said [tex] \lim_{x\rightarrow 0^-}\frac {|x|}{x} = \lim_{x\rightarrow 0^-}\frac {x}{-x} = -1 [/tex] i'd by that, but the text says [tex] \lim_{x\rightarrow 0^-}\frac {|x|}{x} = \lim_{x\rightarrow 0^-}\frac {-x}{x} [/tex]
 
  • #8
You're asking why |x| = -x when x is negative? Isn't that part of its definition?
 
  • #9
yeah that makes sense, but if x is negative then why isn't the denominator negative as well, since it's x? x-->0- holds for both variables in the function |x|/x
 
  • #10
x is negative when x < 0.
 
  • #11
ktpr2 said:
says here the partial limits are +/- 1.

If it said [tex] \lim_{x\rightarrow 0^-}\frac {|x|}{x} = \lim_{x\rightarrow 0^-}\frac {x}{-x} = -1 [/tex] i'd by that, but the text says [tex] \lim_{x\rightarrow 0^-}\frac {|x|}{x} = \lim_{x\rightarrow 0^-}\frac {-x}{x} [/tex]

That's because YOU EDITED YOUR POSTS.You had a square in the denominator... :wink: :rolleyes:

Daniel.
 
  • #12
ktpr2 said:
yeah that makes sense, but if x is negative then why isn't the denominator negative as well, since it's x? x-->0- holds for both variables in the function |x|/x

Let's say that
[tex]x=-5[/tex]
or something like that.
Then we have
[tex]\frac{|x|}{x}=\frac{|(-5)|}{(-5)}=\frac{5}{-5}=\frac{-(-5)}{(-5)}=\frac{-x}{x}[/tex]

Since [itex]x[/itex] is negative, the negation in the numerator makes the numerator positive, and the denominator is negative.
 
  • #13
dextercoiby - yeah that was a typo i removed; changes everything sorry about that.

hurkyl sure -x is negative but how does that force |x|/x to be negative. If x is negative, i'd expect something like -x/-x beause 2) |x| is -x and for x we're taking x -> 0- for denominator x.
 
  • #14
okay. The way I'm looking at this process now is:

|x| gives us -x
---
x gives us plain old x

then we actually take the limit
and we get
-(-x)
----
- x

which gives us

-x/x = -1

is that's what going on?
[edited for conceptual misunderstandings]
 
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  • #15
Just because x is negative doesn't mean you should put a negative sign in front of it.
 
  • #16
ktpr2 said:
dextercoiby - yeah that was a typo i removed; changes everything sorry about that.

hurkyl sure -x is negative but how does that force |x|/x to be negative. If x is negative, i'd expect something like -x/-x beause 2) |x| is -x and for x we're taking x -> 0- for denominator x.

You seem to be confused about the following fact:

If x is negative, -x is not negative.

Take a moment and look at the sentence I just wrote, and make sure you understand it. You appear to be confusing negation (an operation) with something being negative (a property).
 
  • #17
okay i got it now. thanks for all the help Peoples of Physics and Chemistry. (POPAC)
 
  • #18
hmm so -x is negation of x, not negative x, which is the negation of positive x?
eidt - accidentally typed negative when ishouldve used negation
 
  • #19
ktpr2 said:
hmm so -x is negation of x, not negative x, which is the negation of positive x?
eidt - accidentally typed negative when ishouldve used negation

One common misconception is essentially that
-x
is thought of as what would be written as:
-|x|

If that makes any sense at all.
 
  • #20
If x is negative then, yes, |x|= -x when is positive because x itself is a negative number. [itex]\frac{|x|}{x}= \frac{-x}{x}[/itex] which is a positive number over a negative number. That's negative.
 
  • #21
ktpr2, it's really not as complicated as it may appear from this thread.

The definition of |x| is:

[tex]
|x|=\left\{
\begin{array}{cc}
x, & \mbox{if} x \geq 0\\
-x, & \mbox{if} x<0
\end{array}\right.[/tex]

So it's always positive (or zero).
Alternatively: |x| is equal to the distance from the origin and distances are always non-negative.

So |x|/x is positive if x is positive, because you are dividing something positive by something positive.
|x|/x is negative if x is negative, because you are dividing something positive by something negative.

Using the definition, |x|/x is equal to 1 if x is positive and -1 if x is negative.
 

1. What is the definition of the left limit of |x| at x=0?

The left limit of |x| at x=0 is the limit of the function |x| as x approaches 0 from the left side. This means that the values of x are approaching 0 from negative values, and the function is being evaluated at those values.

2. How is the left limit of |x| at x=0 different from the right limit?

The left limit of |x| at x=0 is calculated by approaching 0 from the negative side, while the right limit is calculated by approaching 0 from the positive side. In this case, the left limit is equal to -1, while the right limit is equal to 1.

3. What does the left limit of |x| at x=0 represent graphically?

The left limit of |x| at x=0 represents the value of the function as x approaches 0 from the left side. This can be seen graphically as the y-value of the function at the point where the graph of the function approaches 0 from the left side.

4. How is the concept of continuity related to the left limit of |x| at x=0?

The left limit of |x| at x=0 is a necessary condition for the function |x| to be continuous at x=0. If the left limit and right limit are not equal, then the function is not continuous at that point.

5. Can the left limit of |x| at x=0 be evaluated using the standard limit laws?

Yes, the left limit of |x| at x=0 can be evaluated using the standard limit laws. In this case, the left limit can be calculated by substituting a value slightly less than 0 into the function, such as -0.0001, and evaluating the limit as x approaches this value.

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