At what points are the functions in Exercises 13–30 continuous...

Question

At what points are the functions in Exercises 13–30 continuous?

f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2

3, x = 2

4, 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. Or the function H of Z, defined as H of Z is equal to. Curly bracket of Z 3 minus 27, all divided by Z 2 minus 9, if Z cannot equal plus or minus 3, or if Z equals 3. And 6, if Z equals -3, determine the intervals of continuity. Awesome. So it appears for this particular problem, we're asked to take our function for H of Z. And we're asked to determine the intervals of continuity. So that's what we're ultimately trying to solve for is what are the intervals of continuity. So now that we know what we're solving for, let's read off our multiple choice answers to see what our final answer might be. A is all Z except Z equals 3. B is all Z except Z equals -3, C is all Z, and D is all Z except Z equals 3 and Z equals -3. Awesome. So first off, in order to determine the points of continuity, we need to check where the function H of Z is not defined or has a discontinuity present. So, meaning we have to find the limit at Z equals 3, so we're trying to find the limit at Z equals 3 and Z equals -3. So now we need to calculate the limit as Z approaches -3 of H of Z. Fantastic. So with that said, or with, with that in mind, I should say. Our first step is we need to focus on the left hand limit, and then we need to focus on the right hand limit. So turning our attention to the left hand limit, so that's approaching from the left, we need to write that the limit as Z. Approaches -3 from the left of Z to the power of 3 minus 27, all divided by Z to the power of 2 minus 9, which is going to be equal to. The limit as Z approaches -3 from the left of Z2 + 3 Z. Plus 9, all divided by Z + 3, when we do a little bit of simplifying, and then when we do the direct substitution method, we will receive a result of negative infinity. And for the right hand limit, so approaching from the right now, you need to write that the limit as Z approaches -3 from the right now. Of Z 3 minus 27 divided by Z 2 minus 9, which is going to be equal to the limit as Z approaches -3 from the right of Z2 + 3 Z + 9 all divided by Z + 3, which when we do the direct substitution method, we will receive an answer of positive. Infinity, fantastic. So now we need to note that since the right hand limit and the left hand limit are not equal to each other, the limit of the function at Z equals -3 does not exist. So let's make a note of that. So, Z equals -3 gives us a, does not exist. So the function, once again, the limit of the function at Z equals -3 does not exist. So D and E. Awesome. So moving right along here, so therefore, as a result, the function is not continuous at Z equals -3, so it's DNE and not continuous. That's what NC stands for, and it's not continuous. So moving right along here. So now, our next step is we need to calculate the limit as Z approaches 3 of H of Z. Fantastic. So, with that in mind, what we need to do is we need to take Our limit, so we need to take the limit as once again, let us know that we're taking the limit as Z approaches 3 of the power of 3 minus 27 divided by Z 2 minus 9, which is going to be equal to the limit as Z approaches. 3, so as Z approaches 3 of when we simplify, which is Z 2 + 3, Z + 9, all divided by Z + 3, which we do the direct substitution method, we'll get an answer of 9 divided by 2 or nine halves, and we need to note that this is very important, that since the limit as Z approaches 3 of H of Z cannot equal H of 3, the function will be not continuous, so NC, so not continuous at Drum roll, it's going to be not continuous at Z equals 3. So, therefore, without a doubt, H of Z will be continuous for all Z except Z equals 3 and Z equals -3. So looking at our multiple choice answers, the correct answer without a doubt has to be multiple choice answer letter D. Which once again is all Z except Z equals 3 and Z equals -3. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

At what points are the functions in Exercises 13–30 continuous?
f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2
3, x = 2
4, x = −2

Key Concepts

Continuity of Functions

Limits

Piecewise Functions

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