Limits of an Absolute Value

  • Thread starter smerhej
  • Start date
  • #1
20
0
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim x→−8 [ 8 − |x| OVER 8 + x ]






The attempt at a solution

First → abs(x) = -x ; x<0 / x ; x≥0

Therefore, our equation is going to look like 8 − -x OVER 8 + x

If approaching -8 from the left, I got 8 − -(-8-) OVER 8 + -8-

→ 0- OVER 0- . When dividing 0- over 0- you get infinity, correct?

And doing the limit as x approaches -8 from the right gave me infinity as well, thus giving the answer, that as X approaches -8, Y approaches infinity. But that's wrong. Can someone help please?
 

Answers and Replies

  • #2
ehild
Homework Helper
15,543
1,914
What is 8-(-x)???? Is not it the same as the denominator?

ehild
 
  • #3
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,394
1,045
0 over 0 is not infinity. It's an indeterminate form.

(8-(-x))/(8+x) = ? provided that x ≠ -8
(parentheses are important).
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,847
966
The crucial point is that, for x going to -8, you can assume that x< 0. |x|= -x.
 
  • #5
20
0
Right, so I get the equation (8+x)/(8+x) = ? . Now I'm not entirely sure where to go from here.

Just simply putting the value -8 into the equation gives 0/0 (which is wrong), and I'm not entirely sure how to change the way this equation looks..

Would saying that it equals 1 be fair? Seeing as how the numerator and the denominator are the same?
 
  • #6
35,289
7,140
Right, so I get the equation (8+x)/(8+x) = ? . Now I'm not entirely sure where to go from here.

Just simply putting the value -8 into the equation gives 0/0 (which is wrong), and I'm not entirely sure how to change the way this equation looks..

Would saying that it equals 1 be fair? Seeing as how the numerator and the denominator are the same?
Fair has nothing to do with it. Since the numerator and denominator are the same, for all negative values of x other than -8, the value of the expression (8 + x)/(8 + x) is 1. From this, you should be able to say what the value of the limit is.
 
  • #7
ehild
Homework Helper
15,543
1,914
There is a definition, that a function f(x) has the limit A at x0 if to every sequence xn convergent to x0 the sequence f(xn) converges to A. This way a function can have a limit where it is not defined. Such an example is limx-->1f(x)=(x^2-1)/(x-1). f(x) is not defined at x=1, but for all x≠1 it is equal to x+1, so its limit is 2.

ehild
 
  • #8
20
0
Ah thank you very much! And you are right, fair does have nothing to do with it.. I'll be sure to be more careful with my words.
 

Related Threads on Limits of an Absolute Value

  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
14K
Replies
12
Views
12K
  • Last Post
Replies
11
Views
3K
  • Last Post
Replies
1
Views
978
  • Last Post
Replies
15
Views
2K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
8
Views
941
  • Last Post
Replies
1
Views
4K
Top