Which of the following statements about the function y=f(x) gr...

Question

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

a. limx→2 f(x) does not exist.

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the validity of the following statement based on the given graph of Y equals HX. The limit as X approaches -2 of HX does not exist. Awesome. So we're trying to prove whether it's true or false that this limit as X approaches -2 of HX does not exist, since we're asked to determine the validity of the following statement based on the given graph of Y equals H X. So looking at our graph of Y equals H X, we can see that we have two separate curves that are not connected. Our first curve, it touches the X axis. And it appears to be like a semi-circle or a circle that's been cut in half, and it's like a round parabolic shaped curve almost. And our second curve is V shaped, but it's not quite a true sharp V. It's more like of a curved V. And as you can see from both of our curves, it looks like we might have a couple of holes, and not my, it looks like we do because our White circle, so it's a black border with a white dot in the center, represents a hole, whereas the red dots represent a closed point, whereas the circles open. So, as you can see, we have two separate curves, and now that we have a good visualization, what's happening, we need to note that our final answer is going to be either a true or B, false. So first off, let us note that as X approaches -2 from the left, the function approaches a value of -1. So thus we can go ahead and write that the limit as X approaches -2 from the left. That's what that negative symbol and the power means. It means it's approaching from the left. Of H of X is going to be equal to -1. And as X approaches -2 from the right, the function will approach a value of -1. So therefore we can go ahead and write that the limit as X approaches -2 from the right. That's what that positive symbol in the power means. So, the limit as X approaches -2 from the right of H of X is going to be also equal to -1. So since the limit as X approaches -2 to the power of -1. So, where the limit as X approaches negative 2 from the left, that's what that to the power of negative symbol means. So as we can say that sense, the limit as X approaches negative2 from the left of H of X is equal to the limit as X approaches -2 from the right. is equal to -1. It follows that the limit as X approaches -2. Of H of X. It's going to be equal to -1. So therefore, as a result, the limit does indeed exist and the given statement is false. So our final answer has to be B false. And our answer is false because we're told that the statement originally states in the problem itself that this limit as X approach. 2 of HFX does not exist. However, we just proved that this particular limit does indeed exist. So that means that the statement is false. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

a. limx→2 f(x) does not exist.

Key Concepts

Limits

Continuity

Piecewise Functions

Watch next