Question

Use the given graph to identify all x-values at which the function is discontinuous.

Answer 1

1)

Answer 2

2)

Answer 3

@rizwan_uet can you help?

Answer 4

@RadEn can u help?

Answer 5

Ok let's look at the first one.

If we slide over to \(x=-2\), the function is defined where the black dot is located.
So the function value is -1 ish.

But notice that if we try to move any further to the left, we have to `jump` up to grab the other line.

This is called a `jump discontinuity`. The curve doesn't continue in one straight path. That's bad.

Answer 6

ok

Answer 7

Around \(x=2\), the line curves down to negative infinity on the left, and up towards positive infinity from the right side of 2.

We call this an `infinite discontinuity` or sometimes called an `asymptotic discontinuity`.

Answer 8

Then at \(x=4\) we have a `sharp corner`. It would probably be more accurate to describe it as a `cusp`, but whatever :)

Sharp corners like that are bad! Discontinuity!
I can't remember what type of discontinuity it is though :( hmm

Answer 9

okay how about for graph number two i'll look it up for the word at x = 4

Answer 10

Hmm just like the first graph, it looks like we have a `jump` at \(x=-2\) yes? :)

Answer 11

yes we do

Answer 12

Then we move over to \(x=2\).
There are no problems are the point, only AT that particular point.
It's missing!!!

We call this a `removable discontinuity`.

Answer 13

ok

Answer 14

Then at \(x=4\) we have a discontinuity that we saw in the first graph. Can you recognize it? :) hmm

Answer 15

yes the cusp thingee lol

Answer 16

no no no, not a cusp! :) There is no sharp corner.

What's happening is, the line is going up up up on the right. and down down down on the left.

Answer 17

They draw a `pink line` to show that there is an asymptote located there.
Meaning that our function does not exist in the pink area.

Answer 18

okay

Answer 19

So like in the first graph, when we see an asymptote, we have an `infinite discontinuity`.

Answer 20

ohh okay got it

Answer 21

Here is another form of the `infinite discontinuity` that you might see come up.