Determine the points on the interval (0, 5) at which the follo...

Question

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated.

Accepted Answer

Welcome back, everyone. And the interval 0 15, locate the points where the function F has discontinuities. For each discontinuity indicate which continuity conditions are not met. Here we have a graph showing the function F and its values for A. And for our answer choices, A says A equals 7 and the function is discontinuous because the limit as X approaches A of F of X is not equal to F of A. A equals 9 and the function is discontinuous because the limit does not. Exist and says A equals 15. The function is discontinuous because F of A is undefined where A is in the domain of F. B says A E equals 5. The function is discontinuous because the limit does not exist. A e equals 7. The function is discontinuous because the limit as X approaches A F X is not equal to F of A and A equals 9. The function is discontinuous because FFA is undefined where A is in the domain of F. C says A equals 0. The function is discontinuous because F A is undefined, where A is in the domain of F. A equals 7, the function is discontinuous because the limit as X approaches A of F of X is not equal to F A and A equals 9. The function is discontinuous because the limit does not exist. And D says A equals 5. The function is discontinuous because F of A is undefined, where A is in the domain of F. A equals 7, the function is discontinuous because the limit of F of X as X approaches A is not equal to F A and A equals 9, the function is discontinuous because the limit does not exist. Now what are we supposed to figure out here? Well, we're trying to figure out for each discontinuity on the interval 0 15, what the discontinuity is and why the continuity conditions are not met. Now for the interval open parentheses 0, 15 closed parentheses, that means what is happening at 0 and 15 is not included. So visually we can tell there are discontinuity there, but they are not included in our solution. Instead, we would be interested at the discontinuity where A equals 5, where A equals 7, and where A equals 9. So we've identified the discontinuities. Now why are the continuity conditions not met? Well, let's start at A equals 5. Now notice here at A equals 5, the function is discontinuous because F of A is undefined. Here we have a blank space where A is in the domain of F, OK? So, here we can say that at a well. For A equals 5, OK. FFA is. Undefined. Next, let's look at when A equals 7. What do we notice here? Well, notice that at A equals 7, if we were to think about the limit of F of X, OK, as X approaches A, that is as X approaches 7, notice here that from the left it's approaching 31 and from the right it's approaching 31. So at A equals 7, the limit as X approaches 7 of F of X is equal to 31. Well, sorry. Yes, while F of 7, OK, is where the solid dot is and that is 21. So here the function is discontinuous because the limit as X approaches the value of for F of X is not equal to F of that value. In this case F of A. Now, what about, let me write this here, this is when A equals 7. And what about when A equals 9? We'll notice here for this function F of X as X approaches. Sorry, FX here as the limit of FX as X approaches 9 is equal to 19, while the limit of FFX as X approaches. As X approaches 9 from the right is equal to that would be about 30, right? So here at A equals 9, OK. The limit as X approaches A from the left is not equal to the, sorry, of FFX. OK. is not equal to the limit as x approaches a to the right of F of X. Therefore, that means at A equals 9, the limit does not exist. Therefore, that's why it's discontinuous. Thus, when we go back to our answer choices, that tells us that D is the correct answer. Thanks for watching everyone. I hope this video helped.

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

Key Concepts

Continuity of Functions

Types of Discontinuities

The Continuity Checklist

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