Determine whether the following functions are continuous at a....

Question

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.

f(x)= √x−2; a=1

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with continuity. This problem says check the continuity of the function H of X is equal to the square root of the quantity of X minus 7 at A equal to 3, and apply the continuity checklist for your analysis. We're given two possible choices as our answers. For choice A, we have the function is continuous, and for choice B, we have to know the function is not continuous. Now the question wants us to determine whether this function is continuous. By setting X equal to A, which is going to be equal to 3, and apply our continuity checklist at this point. So, recall your checklist that you need to apply to see if a function is continuous, and we actually have 3 Statements that we need to determine if they're true or not. The first one. Is that F of A? Is defined So the function has to be defined at the point. The second item that we need to check is that the limit. As X approaches A of F of X. Exists. Though the limit As X approaches A of our function has to exist. And then finally, the last condition that we need to check. Is that the limit As X approaches A. Of X sorry, of F of X is equal to. F of A So the limit as X approach A of F of X has to be equal to the value of the function F of A at that point. So, if any of these three conditions fail, then our function is not continuous. So, we're going to start off by checking the first condition that F of A is defined. So, here, we'll be using our function H of X, which we're given a problem, and A is going to be equal to 3. So, we'll be looking for H of 3. And that's going to be equal to the square root of the quantity of 3 minus 7. Which, when we simplify that is equal to a square root of minus 4. Now, we cannot have a negative number underneath the square root sign. So that means that this function is not defined at A equal to 3. That means our first condition actually fails. And I do not need to check the other two conditions, because even if they are true, our function is not going to be continuous. So, since one of them failed, I can easily say that this function is not. Continuous And again, that's because if any of these conditions fail. The function is not going to be continuous. Only if all 3 are met is the function continuous. So now that I have my answer, I can look at my answer choices, and the correct answer choice corresponds to answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.
f(x)= √x−2; a=1

Key Concepts

Continuity of Functions

Limit of a Function

Square Root Function

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