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?

d. limx→c f(x) exists at every point c in (-1,1).

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 H of X. The limit as X approaches C of H of X exists at every point C in parentheses -3-1. Awesome. So it appears that ultimately we're trying to take this given statement which states that the limit as X approaches C of H of X exists at every point C in parentheses -3-1, and once again they're in parenthesis, meaning it's an open interval. Now, if it was a square bracket, it would be a closed interval. And once again we're trying to prove whether or not this statement is true or false based on our graph of Y equals H X. So now that we know what we're ultimately trying to solve for, let's take a moment here to investigate our graph that is provided to us. Now as you can see, we have our curve for Y equals HX represented in red. And as you can see, it looks like it's two separate curves, but it's technically the same curve. The reason why it's two separate curves is because it looks like our first or the curve begins, touches the X axis at what appears to be -3, and then at the point what appears to be possibly negative 1, it looks like there leads to a hole, and a hole means that this curve goes and then it stops at a hole. Like it disappears into the hole and then it starts back up again at about -1.3, and then it goes into what appears to be a V-shaped curve, but it's more like a curvy V like V shape, not a straight true V graph shape because it's more curved, and then it starts from the left at a solid point and then leads to a hole on the right. Whereas initially our curve begins in the semi-circle shape, which is touching the X-axis. So now that we have a visualization of what's happening for this particular prom, let us note that once again we're trying to figure out whether or not the statement is a true or B false. So first off, let us note that for the limit, as X approaches C of H of X to exist at every point C in the open interval parentheses 33, so open interval -3 -1. So in order for this particular limit for the limit as X approaches C H X to exist at every point C within this open interval, the following conditions must be met. Firstly, the function must be defined on the open interval around each point in the open interval, except. At the point itself from the graph we can see that the function is defined on parentheses -3-1. There is a hollow point or a hole at -2-1, but this does not make the function undefined at X equals -2, because there is a solid point at -2.0. Secondly, for each point C in the open interval, both the left hand limit, which is the limit as X approaches C from the left of H of X. And the right hand limit, which is the limit as X approaches C from the right of H of X, must exist and must be equal to each other, and this will ensure that the two-sided limit, which the limit as X approaches C of H of X exists. So we need to note. That 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 of H of X, which is going to be all equal to the limit as X approaches -2 of H of X, which is gonna be all equal to -1. Fantastic. So, in general, for each point C in the given interval, the limit as X approaches C from the left of H of X, which let's write this down. So the limit as X approaches C from the left of H of X is going to be equal to the limit as X approaches C from the right. Of H of X is going to be equal to the limit as X approaches C of H of X, which is going to be all equal to H of C. So therefore, as a result, the function satisfies the second condition. And thirdly, the limit at each point must be finite. So we need to note that there is no point on the given interval at which the function approaches infinity. So therefore, as a result, the function satisfies this specific condition. So, fourthly, lastly, we need to note that, and we also need to confirm rather that the function. Should not have any jump discontinuities at any point within the interval. A jump continuity, so discontinuity where it disconnects occurs when the left hand and right hand limits exist but are not equal to each other. So there is a discontinuity at X equals -2, but this is not a jump discontinuity because the left and right hand limits are equal as discussed in the second condition. So therefore, as a The function meets the 4th condition. So therefore, without a doubt, the given statement is true. So our final answer 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?

d. limx→c f(x) exists at every point c in (-1,1).

Key Concepts

Limits

Continuity

Piecewise Functions

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