Why is the absolute value of x equal to -x for values under zero?

In summary, the conversation discusses the concept of absolute value and how it applies to negative values. The explanation is given that the absolute value of a negative number is the opposite or positive value of that number. The conversation also includes a question about finding the limit of a function using the delta-epsilon method and L'hopital's rule, with the final answer being 2.
  • #1
ne12o
5
0
Here, it says that for the limit f(x) = |x| / x,

|x| = { x, x > 0
-x, x < 0 }

What I don't undestand is why is |x| = -x for values under zero? Isn't the absolute value for negative values just x and not -x?

thanks.

EDIT: I don't want to start a new thread, but I got stuck on this next question :(

Lim x approaching -1 of
3(1-x^2) / x^3 + 1

I tried multiply the equation by x^3 - 1 / x^3 - 1
and I ended up with 0 / -2.

The answer is however 2... any help would be helpful;!

Thanks
 
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  • #2
Example: |-7| = -(-7) = 7.

Do you see why we need the minus sign?
 
  • #3
morphism said:
Example: |-7| = -(-7) = 7.

Do you see why we need the minus sign?

Wow!1
Thanks a lot! I can't believe I cannot see such a simple logic...
 
  • #4
Ofcourse, what you could do is, by knowing in forhand the limit, to use the delta-epsilon method. But that would become quite messy, I think.

Your strategy of multiplying the limit with 1 with apropriate numerator and denominator is clever, and is often useful when you deal with square roots.

Here however, you see that the denominator and numerator have the same limit: 0. That's an indication to use L'hopital. As far as I can see your limit is equal to

[tex]

lim_{x \rightarrow -1} \frac{3-3x^{2}}{1+x^{3}} = lim_{x \rightarrow -1}\frac{-6x}{3x^{2}} = lim_{x \rightarrow -1} \frac{-2}{x} = 2
[/tex]
 
  • #5
haushofer said:
Ofcourse, what you could do is, by knowing in forhand the limit, to use the delta-epsilon method. But that would become quite messy, I think.

Your strategy of multiplying the limit with 1 with apropriate numerator and denominator is clever, and is often useful when you deal with square roots.

Here however, you see that the denominator and numerator have the same limit: 0. That's an indication to use L'hopital. As far as I can see your limit is equal to

[tex]

lim_{x \rightarrow -1} \frac{3-3x^{2}}{1+x^{3}} = lim_{x \rightarrow -1}\frac{-6x}{3x^{2}} = lim_{x \rightarrow -1} \frac{-2}{x} = 2
[/tex]

Thanks for the help! I was stuck on this equation for a while.
It's been about a year since I've even touched Calculus... I'm extremely rusty on the basics, and now I have to relearn everything!
 

1. What is the definition of a limit of absolute value of x?

The limit of absolute value of x is the value that a function approaches as the input variable x gets closer and closer to a specific value, without actually reaching that value. It is denoted by the symbol "lim" and is a fundamental concept in calculus and mathematical analysis.

2. How is the limit of absolute value of x calculated?

The limit of absolute value of x can be calculated using algebraic methods or graphically by analyzing the behavior of the function near the given value of x. It can also be calculated using the epsilon-delta definition, which involves finding a value of delta that corresponds to a given epsilon value.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function approaching the given value of x from one direction, either the left or the right. A two-sided limit, on the other hand, considers the behavior of the function approaching from both the left and the right, and the limit only exists if the one-sided limits from both directions are equal.

4. Can the limit of absolute value of x be undefined?

Yes, the limit of absolute value of x can be undefined if the function has a discontinuity or a vertical asymptote at the given value of x. In these cases, the limit does not exist because the function approaches different values from different directions.

5. How is the limit of absolute value of x used in real-life applications?

The limit of absolute value of x is used in many real-life applications, such as modeling the growth of populations or the decay of radioactive materials. It is also used in engineering and physics to analyze the behavior of systems and to optimize solutions. Additionally, it is used in economics and finance to calculate rates of change and to make predictions.

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