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?

c. limx→1 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 -1 of HX does not exist. So it appears for this particular problem we're asked to determine whether or not this following statement that's based on our given graph of Wals H of X is either true or false. So we're trying to figure out if the limit as X approaches -1 of HX does not exist. Is this true or false for this particular statement? So with that said, let's quickly look at the graph that is provided to us. So we have our Y equals HX function graph, and it appears to have two separate curves, but actually, essentially what's happening here is we have a solid point and then it looks like it leads to a hole, and then another hole for our first part of our curve because it's all part of one curve, but it starts it has a hole, so then it goes, it stops where the hole is and then it starts up again. To form a V shape. So it starts in this semicircle, half a circle, parabolic shaped curve, and then it continues into this what appears to be V shaped, but it's not like a true V graph because it's not a sharp straight lines, rather, it's more of a curved V shape that has a hole on the left and then it leads to a, sorry, it has a solid dot on the on the left and then it leads to a hole on the right. So now that we have a visualization of what's happening here, noting once again that our semicircle part of the curve touches the X-axis, whereas our V-shaped curve is on the Y axis and it intersects at -1. So now that we have a nice visualization of what's happening here, we now need to determine if it's a true or B false. So with that said, we need to note that first off that as X approaches -1 from the left, that will mean that the function approaches a value of 0. So therefore we can state that the limit as X approaches -1 from the left. That's what that negative symbol and the power means, is from the left. So the limit as X approaches -1 from the left of H of X is going to be equal to 0. And similarly, we can state and see that as X approaches -1 from the right of our function. We can see that as it approaches -1 from the right, the function will approach a value of -3. So we can go ahead and write that as X, so the limit as X approaches -1 from the right, that's what that positive symbol means in the power. So the limit as X approaches -1 from the right of H of X is going to be equal to -3. And we can see this on our graph itself. So we need to note that since the limit as X approaches -1 from the left of H of X. Can it cannot equal the limit as X approaches -1 from the right of H of X. It follows that the limit as X approaches -1 of H of X does not exist. So D and E, so it does not exist. So because we just stated that the limit as X approaches -1 from the left cannot equal the limit as X approaches -1 from the right for both of them for H of X, that means that since these two limits cannot equal each other, that means that the limit as X approaches -1 of H of X does not exist as a result. So with that said, without a doubt, the given statement has to be true because indeed, what we just determined together does match the statement that's given to us by the problem itself, so it has to be a true. 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?

c. limx→1 f(x) does not exist.

Key Concepts

Limits

Continuity

Piecewise Functions

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