### Video Transcript

Discuss the continuity of the function ๐ at ๐ฅ equals negative two, given ๐ of ๐ฅ is equal to ๐ฅ cubed plus eight over ๐ฅ squared minus four if ๐ฅ is not equal to negative two and itโs equal to negative three, if ๐ฅ is equal to negative two.

So there are four options here and Iโm going to start with B which is that the function is continuous at ๐ฅ equals negative two. And the other three options are saying that the function is discontinuous for a variety of reasons. So A says itโs discontinuous at that point because the value of the function ๐ of negative two is not equal to the limit of that function as ๐ฅ tends to negative two. Option C says that ๐ of negative two is undefined. And option D says itโs discontinuous because the limit doesnโt exist.

These four options give four different scenarios. And I think itโs worth thinking about what the graph of ๐ looks like near ๐ฅ equals negative two for each of them. So in option A, limit as ๐ฅ tends to negative two if ๐ of ๐ฅ does exist, but it doesnโt equal ๐ of negative two. And we can see this scenario being illustrated in the diagram in the bottom, left-hand corner.

You can see that the limit as ๐ฅ tends to negative two does exist. Iโll pour it on; there it is; itโs somewhere maybe around minus 1.5. But unfortunately, it doesnโt match the value of the function when ๐ฅ is negative two, which is minus three. So thatโs one thing that could go wrong, one thing that could cause the function to be discontinuous.

Option B is very similar, except the limit as ๐ฅ tends to negative two of ๐ of ๐ฅ is now negative three. And so that point is kind of in the right place to fill the gap in that curve and the function is continuous.

Option C is what happens when you donโt define ๐ of negative two. I think itโs pretty clear that we have to find ๐ of negative two; in fact, itโs here. So we know that ๐ of ๐ฅ is negative three if ๐ฅ is negative two, but had we not done that itโs possible that this thing here is not defined and so the function is not defined.

And finally, option D where the limit as ๐ฅ tends to negative two, ๐ of ๐ฅ does not exist. And you can see in the diagram Iโve suggested that could be because thereโs a vertical asymptote and ๐ฅ equals negative two. And so the function is discontinuous there even though it is defined. ๐ of negative two is equal to negative three, but either side of that the function is going off to plus one negative โ. Of course, it doesnโt have to be an asymptote. It could just be at a jump gap, discontinuity. Either way, itโs an option.

Step one is to find the limit as ๐ฅ tends to negative two of ๐ of ๐ฅ. Why do we do that? Well, if the limit exists, but is not equal to ๐ of negative two, which we said was negative three, then we know the answer is option A. If instead the limit does exist, but is equal to negative three, then the function is continuous. And so option B is our answer. Option C Iโve said we can safely discount because ๐ of negative two is defined; itโs negative three. And if the limit doesnโt exist, then our answer is option D.

We want to find this limit as ๐ฅ tends to negative two and as ๐ฅ tends to negative two is never actually negative two. And so the value of this limit doesnโt depend on the value of the function when ๐ฅ is equal to negative two; it only depends on the value of the function when ๐ฅ is not equal to negative two. And so we can safely use this formula in place of ๐ of ๐ฅ. So we got this rational function here: ๐ฅ cubed plus eight over ๐ฅ squared minus four.

And we know that for a rational function, itโs continuous wherever itโs defined. So we could just try substituting in negative two to this rational function. Hope itโs defined there. And if it is, then our value will also be our limit by the definition of continuity. Okay, so we substitute negative two in. And because negative two cubed is negative eight and because negative two squared is four, we get zero over zero which is indeterminate. So thereโs no luck there; we canโt tell what the limit is from that. But what we can tell from the factor theorem is that both the numerator and denominator of this rational function have a factor of ๐ฅ plus two. So letโs divide that out.

So you might be able to recognize that the denominator is the difference of two squares. And so this factor here is going to be ๐ฅ minus two. What about the other factor of the numerator? Well, it looks like it should be a quadratic. So we could write down the general form of a quadratic and solve the coefficients. Okay, thatโs the general form of a quadratic. Let me now multiply out the brackets in the numerator.

Iโve multiplied out the brackets on the numerator. And of course, I want to end up with ๐ฅ cubed plus eight, which it is after all what weโre trying to factorize. And now I will compare coefficients. The coefficient of ๐ฅ cubed on the left is ๐ and on the right itโs kind of hidden; itโs one. So I can say that ๐ is equal to one. The coefficient of ๐ฅ squared on the left is two ๐ plus ๐ or as we know ๐ is one; itโs two plus ๐. And on the right, itโs zero. Thereโs no ๐ฅ squared term, which is explicitly written out. So we see that two ๐ plus ๐ equals zero. As we said before, ๐ is one, so two plus ๐ equals zero and ๐ is equal to negative two.

Itโs a very similar story for the coefficient of ๐ฅ, which is two ๐ plus ๐ on the left and well not mentioned on the right zero. Solving that gives ๐ equal to four and then our last equation the coefficient of ๐ฅ to the power of zero or the constant term on the left is two ๐, on the right is eight, two ๐ is eight; that agrees with what we have, ๐ is four. And so weโve got consistency, which is good, and our coefficients. So letโs put them in. Okay, there we go.

And now Iโm going to claim that I can cancel these two factors of ๐ฅ plus two. Why can I do that? Well, Iโm saying that neither of these is zero because ๐ฅ is not equal to negative two because weโre taking the limit as ๐ฅ tends to negative two, which means explicitly avoiding values of ๐ฅ, which are negative two.

Okay, so weโve got the limit as ๐ฅ tends to negative two of a slightly simpler rational expression, which we can think of as a rational function. And weโre going to think about evaluating it at negative two as we tried before. And this time, weโre going to hope that we donโt get zero over zero. Indeterminate thing we hope that weโre going to get a proper real number because if we get a real number, when we evaluate the expression we know that it will also be the limit of that expression as ๐ฅ tends to negative two.

So here, Iโve just substituted in negative two. Now, letโs evaluate. So we simplify the numerator to find itโs 12 and the denominator to find itโs negative four. And 12 divided by negative four is negative three. And because itโs not indeterminate or undefined, we know that thatโs our limit. Okay, great! Thatโs step one completed. What do we do now?

Letโs have another look at the options. Remember we already discounted option C coz we knew that ๐ of negative two is defined. Now, we found that the limit of ๐ of ๐ฅ as ๐ฅ turns to negative two is negative three. We know that limit of ๐ of ๐ฅ does exist. And so this option D canโt be true, so thatโs also eliminated as an option.

So this leaves two options: option A and option B. And as discussed before, this hinged on whether ๐ of negative two was equal to limit of ๐ of ๐ฅ as ๐ฅ tends to negative two. Well of course, we can see what ๐ of negative two is quite clearly. Weโre told itโs negative three.

And so comparing that to the limit as ๐ฅ tends to negative two of ๐ of ๐ฅ, we see that theyโre the same. So that eliminates option A, which says that theyโre different and not equal, where in fact they are equal to negative three. So the only option that weโre left with, and of course the option that makes sense, is that the function is continuous at ๐ฅ equals negative three.

And it makes sense of course because thatโs the definition of continuity of a function at a point. For function to be continuous at a point, it has to be defined at that point. And more than that, the definition of the function at that point has to be equal to the limit of the function as ๐ฅ approaches that point. So our answer is B. The function is continuous at ๐ฅ equals negative two.