Question

AP Calculus: Describe the discontinuities. Are there vertical asymptotes? f(x)=(x^2-9)/(x^2-2x-15)

Answer 1

why would x=3 not be a non-removable

Answer 2

because x-3 is a factor of both numerator and denominator

Answer 3

and + its on the numerator is that why?

Answer 4

sorry : (x+3)(x-3)/(x-5)(x+3)
so x=5 is the only vertical asymptote

Answer 5

because why?

Answer 6

vertical asymptotes are defined through the limit. If the lim approaches infinity or negative infinity when x--> c then x=c is a vertical asymptote. As you can check if we take a limit as x--> -3, the result is not infinity . So x=-3 is not a vertical asymptote
Meanwhile as x--- > 5 (from the right), the limit is +infinity, so x=5 is the vertical asymptote.

Answer 7

non-removable rite for x=5

Answer 8

my teacher uses non-removable and removable

Answer 9

removable or non removable is not an indication of the existence of vertical asymptote. There is an example where the discontinuity is nonremovable but it doesn't have asymptote there.

Answer 10

my teacher put only x=5 for non-removable why didn't he put x=3 since the (x-3) in the numerator didn't cancel out

Answer 11

Oops

Answer 12

@Hero same thing with x=3 rite? :)

Answer 13

it's \[\frac{ (x-3) }{ (x-5) }\]

Answer 14

Yeah, I know. I just said I made a mistake

Answer 15

But hopefully, you get the idea.

Answer 16

\[\frac{ (x+3) (x-3) }{ (x+3) (x-5) }\]

Answer 17

anytime you can cancel common factors from the numerator and denominator, that essentially creates a "hole" in the graph... a discontinuity (removable).

Answer 18

That's not what creates the hole in the graph.

Answer 19

yeah i know so wouldn't x=3 also be a non-removable like (x-5) x=5

Answer 20

so tell me what does...???

Answer 21

No, x = 5 is what is non-removeable

Answer 22

@dpalnc, think about it, the hole is still there even if you don't cancel anything. The fact that the denominator cannot equal zero is what creates the hole.

Answer 23

why not x=3 as well? :\ it didn't cancel out with anything like (x-5)

Answer 24

i guess we have different definitions of a "hole" then....

Answer 25

I would suggest that you start from definition of continuity and of vertical asymptote.

Answer 26

@lilfayfay, x = 3 does not cancel because (x - 3) is in the numerator

Answer 27

x = 3 does not create a hole

Answer 28

oh so since it's in the numerator it can't be in the "non-removables" like (x-5) was in the denominator? xD

Answer 29

ok gotcha ;)

Answer 30

"Cancelling common factors from the numerator and denominator creates holes in graphs"

FALSE

Answer 31

@Hero so since (x-3) is in the numerator it can't be a part of the "non-removables"?

Answer 32

precisely, since it does not create a hole to begin with. Only certain things can create "holes" or "discontinuities"

1. The variable in the denominator of a fraction
2. A square root
3. Absolute Value
4. Piecewise functions

Answer 33

so hero, you'd say that there's a hole at x=-5 in f(x)= \(\large \frac{1}{x+5} \) ???

Answer 34

okay ty, all i wanted to hear was that u agrred with me -phew xD

Answer 35

@dpaInc

Answer 36

correct

Answer 37

so we do have different definitions.... my definition of a hole is that it is only removable...

Answer 38

That's a very limited definition of hole

Answer 39

Especially since that hole is the ONLY hole

Answer 40

well... maybe the text i read it in is limited...

Answer 41

I think you should get rid of the idea of holes and start replacing it with "discontinuity" Because that's the proper term to use to describe such concepts.

Answer 42

then so should a lot of textbooks...

Answer 43

Tell me the name of the textbook you were reading from bro. I have all of them

Answer 44

Page number as well.

Answer 45

well for one I used hostetler/edwards calculus... i think they use that term even up to the last edition....

Answer 46

I want to see the part where it specifically states: "Cancelling common factors from the numerator and denominator creates holes in graphs"

Answer 47

I think hole is an ok term for removable discontinuity

Answer 48

Hey watchmath, do you agree that "Cancelling common factors from the numerator and denominator creates holes in graphs"

Answer 49

That's what @dpalnc said

Answer 50

no :)

Answer 51

Hero....
larson/hostetler/edwards alternate 6th edition.....
page 63, example 1 Cancellation Technique...

Answer 52

It states that cancelling creates holes in graphs? Or did you just mis-interpret what was written?

Answer 53

I have that book bro

Answer 54

The book refers to discontinuities as "discontinuties that are either removeable or non-removeable". Doesn't say anything about holes, however.

Answer 55

lol...
i made u get out ur book???

Answer 56

ok.. it just says the two graphs coincide but one has a hole in it....
and if you notice, the hole occurs at the x value for where the canceled factor is zero

Answer 57

I was just looking for where it describes a discontinuity as a hole. I was particularly interested in that, but was greatly disappointed.

Answer 58

Yes, you misinterpreted it bro. The hole was there even before cancelling.

Answer 59

The hole wasn't created as a result of cancelling.

Answer 60

well, like you said before, you consider a "hole" to be any discontinuity where i consider it to be a removable discontinuity.

to each his own....
whatta u say we leave it at that....

Answer 61

No, I don't consider that. What I'm saying is to disregard the word "hole" altogether because it should not be used to describe discontinuities.

Answer 62

ok.. you do what you want....
i'm over it... good day to you....

Answer 63

The proper words to use are "removeable discontinuity" or "non-removeable discontinuity".

Answer 64

@Hero @watchmath can u help me with another problem?

Answer 65

@lilfayfay, close this question, then post your next question in the open question section.