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?

b. limx→2 f(x)=2

Accepted Answer

Hello 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 H X is equal to 0. So it appears for this particular problem, we're asked to determine the valididity of this following statement, which states that the limit as X approaches -2 of H of X is equal to 0. So we're trying to prove whether or not this statement is true or false, and this. Is going to be based on the given graph of Y equals H of X. So looking at our given graph of Y equals H of X, we could see that we have two separate curves. One curve touches the X axion, and it appears to be like parabolic shaped, and it's like a it resembles half of a circle, a semi-circle, and we also looks like it appears to be two holes. That are along our curve, which are denoted by these white dots with a black circle going around them. So like a white circle with a black line going around them. We also have a second dairy part of our curve, which is separate, which is V shaped, but it's not a true straight V shape, it's more like a curved V shape that intersects the Y axis at -1. Awesome. So as you can see, we have our two separate like two separate curves that are represented in red for Y equals H of X. So now that we have a visualization of what's happening in this particular prong, let us know that we're trying to figure out whether or not the statement is a true or B false. So, let us note first off that as X approaches -2 from the left, that will mean that the function approaches a value of -1. So we can say that the limit as X approaches -2. From the left, that's what that negative symbol and the power means. So as the limit, as X approaches -2 from the left of H of X is gonna be equal to -1. And we can state that as X approaches -2 from the right, the function will approach a value of -1, so 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 as the limit of X approaches -2 from the right of H of X is also going to be equal to -1. And since the limit as X approaches -2 from the left of H of X is equal to the limit as X approaches -2 from the right. And we can say of H of X. We can also say that this is all equal to -1. We can therefore, as a result, simply state that as X approaches. -2, so you say that the limit as X approaches -2 of H of X is gonna be equal to -1. So, because we have just proven that our limit does indeed exist and our limit does exist at -1, that means when we look at our given statement, the limit as X approaches -2 of H X equals 0. We just proved that the limit as X approaches -2 of H of X does not equal 0, it equals -1. So therefore, our final answer, without a doubt, has to be be false. The given 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?

b. limx→2 f(x)=2

Key Concepts

Limits

Continuity

Graph Interpretation

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