Question

What Graphical feature does each discontinuity correspond to on (x−6)/ (|x|−6) ?

Answer 1

The equation is \[(x-6)/\left| x \right|-6\]

I found the small x value to be -6, and the large one 6.

What does each represent on the graph of the function?

A hole, jump, or vertical asymptote?

Answer 2

Does graphing it help at all?

Answer 3

so you are dealing with limits?

Answer 4

I believe so. Teacher hasn't phrased it that way though.

Answer 5

ok. so are you in precalc? and are we looking at x==/-infinity and around x=0?

Answer 6

Sort of. And yes.

Answer 7

ok. good enough. let us say that x= 1,000,000. what does this thing look like when x is huge and positive?

Answer 8

although, according to the problem, it looks like there is only one discontinuity, and that would be around x=0.

Answer 9

I am not sure why there are two values then. Does that make sense?

Answer 10

explain two values.

Answer 11

The positive and negative 6. They both make the denominator equal zero.

Answer 12

it is not the numerator that you are interested in, but the denominator that provides clues to discontinuity.

Answer 13

it is the absolute value of x that we are interested in and what the graph does as x approaches 0 from the right and from the left.

Answer 14

Yes. I understand that part

Answer 15

ok. then as x approaches 0 from the right, the absolute value stays positive and the whole expression decreases without bound. do you see that?

Answer 16

I think so

Answer 17

ok. it is a little more time consuming, but..... take your calculator and substitute x=.1 into the equation. then try x=.01 and see what happens. then try x=.000001.

Answer 18

Am I just evaluating, or graphing?

Answer 19

just evaluating.

Answer 20

They all equal the same value

Answer 21

Equal to 1

Answer 22

i think there is an order of operations problem

Answer 23

you should get about -60 for x=.1

Answer 24

then about -600 for x=.01

Answer 25

Odd

Answer 26

I am not sure what I am doing wrong then...

Answer 27

let me ask you something.

Answer 28

the above equation. does it have any parentheses? that would make a tremendous difference

Answer 29

as it is written, you would do the fraction first and then subtract the 6

Answer 30

Well, there are parentheses around \[\left| x \right| -6\]

My mistake

Answer 31

ahhhh. ok. still not a problem.

Answer 32

so now we are interested in the interval around x=6, are we not?

Answer 33

Yes. Should have noticed that earlier

Answer 34

so, when x> or = to 6, what do you think happens?

Answer 35

we are also going to have to look at x=0, if i am not mistaken

Answer 36

If it is 6, the denominator is equal to zero.

Answer 37

And yes.

Answer 38

ok, i should have said as x approaches 6 from the right. lol

Answer 39

so what happens when x approaches 6 from the left and from the right?

Answer 40

is there a difference or does the value of the fraction remain the same? and if so, what might that value be?

Answer 41

I honestly am not sure.

Answer 42

it is ok to not know.

Answer 43

I can't visualize it for some reason

Answer 44

otherwise you wouldnt be here, right?

Answer 45

I feel like I should though, but yes.

Answer 46

so, tell me, what is the value of |6.1| and |5.9|? please humor me here. lol

Answer 47

They should be the same?

Answer 48

well, since they are both greater than 0, they are both positive, right?

Answer 49

Yes

Answer 50

what i am getting at, is that since they are both greater than 0, so is |x| at x=6.

Answer 51

that means that your fraction is then =(x-6)/(x-6) because |x|=x for all x greater than or equal to 0 (the def. of abs value).

Answer 52

Okay...

Answer 53

stop me at any time. questions are good

Answer 54

So the absolute value sign becomes unnecessary at that point?

Answer 55

only because we are interested in values right now around x=6, which is greater than 0.
so, |x| is x throughout the interval around 6. do you see this?

Answer 56

Yes

Answer 57

ok.

Answer 58

so now, what is the value of the fraction around x=6, assuming that x does not = 6.

Answer 59

it is hard to piecemeal this together. maybe this will help. |x|=x whenever x> or =0.
substitute this into the expression to get (x-6)/(x-6). with absolute values, you need to break the expression into pieces when x is pos and then when x is negative. when x=6, the absolute value of 6 is pos., so we can eliminate the abs value signs. Around x=0, it is a different matter.

Answer 60

so now that you see it all, what is the value of the expression above?

Answer 61

so, (x-6)/(x-6)=1

Answer 62

so this corresponds to a point discontinuity, since the denominator cannot =0

Answer 63

around x=0, the limit does not exist