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?

g. limx→1 f(x) does not exist.

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says, determine the validity of the following statement based on the given graph of Y is equal to GFX. And the statement that we need to look at is the limit, as X approaches 0 of GFX does not exist. Below the problem we're given a graph that has our function G of X plotted on it, and it's a piecewise function. The first piece is a linear piece that goes from the point of -1.0 to the point of 0. -1, and the two endpoints are included on that interval. The next segment is also a linear segment and begins at the open point of 0.1 and goes to the open point of 1.0. We also have a singular point located at 1. -1. And the last segment is also linear, and it goes from the open point of 1.0 to the closed point of 2.1. We're also given two possible choices as our answers. For choice A we have true, and for choice B, we have false. So we need to determine the validity of the statement that the limit as X approaches 0 of G of X does not exist. Now, we call that for a limit to exist, both the left-handed and right-handed limits must approach the same value. So we'll start off by looking at the left-handed limit. So we'll have the limit as X approaches 0 from the left of G of X. Now to determine this value, I'm going to come over to my graph, and I'm going to start at a point that's just to the left of X equal to 0, and I'm going to follow the line for my function G of X until I reach X equal to 0. And as I approach X equal to 0, I see that our function G of X is approaching a value of -1 from the left. So this limit is going to be equal to -1. Next, we're going to look at the limit. As X approaches 0 from the right for a right-hand limit of G of X. And here again, we'll start at a value of access just to the right of X equal to 0, and we're going to follow our function G of X. As X approaches 0 and see what value it approaches, and as X approaches 0, our function G of X is going to be approaching a value of 1 from the right. So this limit is going to be equal to 1. And what we see here is that these two limits are not equal. The limit as X approaches 0 from the left of G of X. is not equal to the limit as X approaches 0 from the right of G of X. Since -1 is not equal to 1. And since these two limits are not equal to each other, that means that the limit as X approaches 0 of G of X does not exist. So that means that our statement here is actually true. And so that is the answer that we are looking for for this problem. And if we come down here to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

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

g. limx→1 f(x) does not exist.

Key Concepts

Limits

Continuity

Graphical Analysis

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