Limit of Absolute Values: Understanding the Concept

In summary, the conversation discusses finding the limit of absolute values and how to evaluate it. The limit is found to be -1 by considering the numerator and denominator separately and simplifying the expression. L'Hopital's rule is not used in the solution.
  • #1
azncocoluver
22
0

Homework Statement


http://www4c.wolframalpha.com/Calculate/MSP/MSP621a026b7befdf1f1d00003b707661e1i20i4e?MSPStoreType=image/gif&s=55&w=126&h=38 [Broken]

Homework Equations


limit x->3- (3x-9)/((abs(3x-9))

The Attempt at a Solution


I don't understand how to find the limit of absolute values..
 
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  • #2
what is the sign of |3x-9| for x<3?
 
  • #3
It would be -|3x-9| = -3x-9.. right?
 
  • #4
the sign is negative - however what you have written is in correct

teh way to do it is as follows
x<3
3x<9
3x-9<0

so it is negative, to make positive let's multiply by -1.

So for x<3
|3x-9| = -(3x-9) = 9-3x
 
  • #5
however an even simpler way would be to do a subtitution (maybe, depending what you're comfortable with...)

so you could let
[tex] y=3x-9[/tex]

then
[tex] \lim_{x \to 3^-} \ \implies \ \lim_{y \to 0^-} [/tex]

and the limit becomes
[tex] \lim_{x \to 3^-} \frac{3x-9}{|3x-9|} \ \implies \ \lim_{y \to 0^-} \frac{y}{|y|} [/tex]
 
  • #6
lanedance said:
the sign is negative - however what you have written is in correct

teh way to do it is as follows
x<3
3x<9
3x-9<0

so it is negative, to make positive let's multiply by -1.

So for x<3
|3x-9| = -(3x-9) = 9-3x

Okay so the fraction is now 3x -9 / 9-3x? How do I get the answer of -1 though? Also, is the limit of y/abs(y) is always going to be -1?
 
  • #7
1) how would you normally evaluate a limit, or where are you stuck. Is this one indeterminate?

2) No. Maybe from the negative side, but that's what we're trying to show isn't it.

Try out evaluating the direct limit first rather than the substitution, we can come back to that, but it is a helpful comparison
 
  • #8
Can you show me the steps to complete this problem because I have no idea where to start.
 
  • #9
well let's star my considering the numerator and denominator separately, qualitatively, what happens to each x goes to 3 from the left

the do you know about l'hopitals rule?
 
  • #10
We can't use L'Hopital's rule yet because my teacher hasn't taught it. Um, the numerator goes to infinity and so does the denominator. So infinity/infinity?
 
  • #11
not quite, have another think, what happens to 3x-9 when x->3

now this doesn't really need L'Hop as we have

[tex] \lim_{x \to 3^-} \frac{3x-9}{|3x-9|} = \lim_{x \to 3^-} - \frac{3x-9}{3x-9}= - (\lim_{x \to 3^-} \frac{3x-9}{3x-9}) [/tex]

as they're identical, i don't think it s a leap to say the final limit is -1
 
  • #12
Oh okay I think I finally understand. 3x-9 / 3x-9 is 1 and since there is a negative outside the parantheses, the limit is just -1 because you can't plug 3 into x since there is no x left.
 

What is the concept of limit of absolute value?

The limit of absolute value is a mathematical concept that determines the behavior of a function as the input values approach a certain point. It is used to find the value that a function approaches as its input values get closer and closer to a specific number.

How is the limit of absolute value calculated?

The limit of absolute value is calculated by finding the value of the function at the specified point and then checking the values of the function as the input values approach the specified point from both the left and right sides. If the values approach the same number from both sides, then that number is the limit of the absolute value function.

What is the significance of limit of absolute value in mathematics?

The concept of limit of absolute value is important in mathematics because it allows us to analyze the behavior of a function and make predictions about its values. It also helps us to understand the continuity and differentiability of a function at a specific point.

Can the limit of absolute value exist if the function is not defined at the specified point?

No, the limit of absolute value cannot exist if the function is not defined at the specified point. This is because the function must be defined and have a specific value at the specified point in order for the limit to exist.

How does the limit of absolute value differ from the limit of a regular function?

The main difference between the limit of absolute value and the limit of a regular function is that the absolute value function will always approach a positive value, regardless of the sign of the input values. This is because the absolute value function always returns a positive value, while a regular function can return both positive and negative values.

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