Question

I don't know how to find discontinuities from a graph. Can you guys help me with this?

Answer 1

I know there is a removable discontinuity at g(x)=1

Answer 2

At x = 1 the discontinuity is not removable.
At x = 5 there is a removable discontinuity.

Answer 3

Is the point x=4 continuous?

Answer 4

Yes.

Answer 5

At x=1 there is a jump so that is a removable discontinuity.

Answer 6

At x=2 the function is continuous.

Answer 7

At x = 1, the limit does not exist. So it is not a removable discontinuity.
At x = 5, the limit exists but is not equal to g(5). This is a removable discontinuity.

Answer 8

x=1 discontinuous
x=2 continuous
x=4 continuous
x=5 discontinuous
Why is it discontinuous at 3?

Answer 9

At x=5 there is a removable discontinuity and that is the only one.

Answer 10

The function is not defined or discontinuous at x = 1, 2, 3 and 5.

Answer 11

so what does that mean?

Answer 12

What does what mean? You can see in the graph that you cannot determine the values of the function at x = 1, 2, 3 and 5. So the function is discontinuous at those points.

Answer 13

There is a vertical asymptote at x=3 because the limit as x approaches 3 from the positive side approaches +infinity and the limit as x approaches 3 from the negative side approaches -infinity.

Answer 14

Yes. And that is also the reason the function is discontinuous at x = 3 because the left hand limit is not equal to the right hand limit.

Answer 15

So do you know how they want me to evaluate the limit as x approaches 5 of g(x)? Do they just want me to tell them that it is discontinuous?

Answer 16

At x = 5, the function is discontinuous but the limit exists!
A limit can exist even though the function may not be defined there.

Answer 17

Oh okay so the limit as x approaches 5=-1

Answer 18

No. The limit as x approaches 5 from the left and the right are both +1.
But f(5) = -1.
The limit exists at x = 5 but does not equal f(5) and hence it is a removable discontinuity.

Answer 19

Okay so f(5)=-1 but the limit is +1

Answer 20

For the limit from the left, start a little bit to the left of x = 5 and locate the point on the graph. As x approaches 5 you can see the limit is approaching +1. Do the same from the right and you can see the limit approaches +1. But the function is -1 at x = 5.

Answer 21

That makes more sense.

Answer 22

Alright.

Answer 23

Thank you for helping me understand it a little better.

Answer 24

You are welcome.