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. On the interval from 0 to 15, locate the points where the function F has discontinuities. For each discontinuity, indicate which continuity conditions are not met. So let's recall three simple conditions for a function to be continuous. Number 1, that function must be defined at point A, right? F of A must exist. Number 2, the limit of F of access X approaches A must exist. And 3, that limit must be equal to the value of the function at a point. So now when we consider the given interval from 0 to 15. The first point that we want to consider is X equals 2, right? Because this is where we have a vertical asymptote. Now, notice that this is a vertical asymptote, meaning at point A equals 2. The function is not defined, right? So F of 2. Does not exist. Because the value of the function at 0.2 does not exist, this means that A equals 2 is a point of discontinuity, and that's our first answer. Now let's consider the function as we can see from 2 onwards. We are within a smooth curve until we approach a point X equals 4, and this is where we have a jump, right? So now let's consider a equals 4. And what we notice is that the value of the function F of 4 exists. It is equal to 10, so we're not violating the first condition. However, the limit does not exist because if we are approaching 4 from the left, right? We would have a limit of 5, and if we are approaching 4 from the right, the limit is going to be equal to 10. So since limit. As x approaches 4 from the left of F of X is equal to 5 and limit as x approaches 4 from the right of F of X is equal to 10, these are not equal to each other. They must be equal to each other to make sure that the limit itself at that point exists. And since they are not equal to each other, well, we can say that we are violating the second condition. So A equals 4 is a point of discontinuity because limit as X approaches 4 from the left of F of X. is not equal to the limit as x approaches 4 from the right of that same function, and we have our second point of discontinuity. So we had our first one, now we have our second one and let's continue. Now we notice that there is another point where we have a break, right? We have a hollow point and a solid point at X equals 7. Notice that the value of the function F at 7 is going to be equal to 12. That's because we're considering the solid point, but the limit at that point, even though it exists, because from the left and from the right we are approaching the same value of 14, that limit exists, right? So limit as x approaches 7 of F of X is equal to 14. However, the function is not continuous because we are violating the third condition. So we are going to say that A equals 7 is the third point of discontinuity, and the reason is because limit. As x approaches 7 of F of X is not equal to F of 7. We're violating the third condition of continuity. And then let's see if we have any additional points. Well, essentially those are just 3 points because from 7 onwards we have a smooth curve and that's it. We can say that there are 3 points of discontinuity and that's it. Thank you for watching.

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 Limit Concept

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