what's up my preo people I'm Michael prj and welcome to part two of AP precalculus topics 1.9 and 1.10 covering vertical oopes and holes of rational functions now in part one we talked about how to identify vertical oopes and holes in rational functions and in this video we're going to focus on the behavior of the function as we get really really really close to a vertical oope or a hole now expressing that behavior is really important because we're going to use something new called limit notation to do so so let's see what we got to do right now so here's the deal if a value xal a is excluded from the domain because it makes denominator zero then that means at that particular value a nothing happens which means F of a is well undefined there's nothing there in that nothingness it's called a discontinuity there's nothing there the graph is not continuous at that particular x value okay so we can't explore what happened at xal a but we can't explore the behavior of the function as we get really really close to that xal a value and what's cool is we can explore what happens is we get really really close from the left of that value and the right of that value so when we think about this we think about getting near that value again what happens when we get to that value well nothing it's going to be a vertical aope or a hole and there's nothing there but what we want to focus on in this video is what happens as we get near like what happens when we get in the neighborhood or close to that value now to do this we use limit notation because that's exactly what limits allow us to do they allow us to explore what happens as we get really really close to an x value on a function without actually getting there so when we talk about the limit as X approaches a what we're talking about is what's happening to the function as we get really really close to a but as I mentioned we could explore what happens is we get really really close to a from the left side or the right side so when we're doing this we can put a little Nega ative in the subscript after the a to determine that we're looking at the left side or we could put a little positive in the subscript after the a to talk about as we're looking at the right side let me explain a little bit by looking at these examples so in this particular graph what we see is a vertical asint right we see a vertical asint at xal 3 now we don't have the graph there we just have the vertical asint but what I want to address is that we could talk about what happens as we get close to this vertical ascope by looking at the right side as X approaches three from the right that little plus sign we see there in the subscript is referencing from the right and as we approach a limit or as we approach a vertical asint from the right we could do one of two things we could go up towards Infinity forever or down towards negative Infinity forever when we hit an ASM toote what we do is we hug it right we get really really close to it without ever actually getting there so that's the idea when we approach from the right we could go up forever or down forever and the same thing for the limit of that function as we approach three from the left we put a little negative on that subscript to represent that we're coming from the left hand side and when we approach a vertical ascope from the left we could go up towards infinity or down towards negative Infinity so the two limit options are either Infinity or negative Infinity as we approach a vertical asint from either side now in this graph we see a hole we clearly see a hole that's located at xal 3 so instead of a vertical ASM at 3 there's a hole but the same thing we could talk about what happens is we approach this hole from the right hand side with a plus sign or a uh from the left hand side with the minus sign so here's the deal near a vertical ASM xal A of a rational function the values of the polinomial function in the denominator are arbitrarily close to zero so the values of the rational function R ofx increase or decrease without bound the corresponding mathematical notation is using limits so we talk about the limit of our rational function as X approaches a from the right could be either positive or negative infinity and the limit of our rational function as we approach that vertical isope a from the leftand side is also positive or negative Infinity the idea is that when you have that value in the denominator that's creating the vertical ascope or that factor as you select values close to a from either side the denominator is getting really really really really small and if the denominator is getting really really really small that means the overall fraction the rational value is going to be getting really really big in a positive or A negative Direction now that may seem a little bit confusing but once we look at some examples I think it'll make a lot more sense but again the main thing I want you to understand is the limit as we approach a vertical ascope on either side is going to be infinity or negative infinity and here's an actual example of that what we have here is a vertical oope at xal 5 because ne5 makes only denominator zero and in the picture we can actually see when we approach that vertical isope from the right side we're going up towards Infinity as we approach from the the left side we're going down towards negative Infinity so again just to try to make sense to this if you pick a number really really close to five from or from to neg5 from the right like let's say -4.9 n999 that means the denominator is going to be a very small number when you divide by a very small number the overall value of the fraction actually gets bigger and it's going to go towards POS infinity or from the left hand side it's going to go towards negative Infinity so again it's really easy to see what's happening as we approach a vertical through a graph what's new is how to represent that using this limit notation now when it comes to holes if the graph of a rational function R has a hole at xals C then the location of the hole could be determined by examining the output values corresponding to input values sufficiently close to C and again I know that could be very confusing but what we're saying is if we think about values close to C from the left or the right then we can figure out what's happening for that now I actually already taught you back in in part one of this um section about how to find the location of the hole meaning how to find the Y value of the hole and I'm calling that L here it's all about reducing away the factors that created the hole and then plugging that value of where the hole is into what's left over if you don't know how to do that go back and watch part one this video it'll definitely explain it so the idea is at that particular location C comma L again x value y value but there's nothing there it's a whole but we approach both sides of that hole we are getting closer and closer to that L value which is the Y value of the hole's location so the limit of our rational function as we approach that hole from the left or from the right is both going to be L the location of the hole now way easier to understand that with a graph so let's look at this problem right here but we actually noticed that this graph actually has a vertical asob as well so we might as well talk about that real quick we see there's a vertical asob at4 because it makes only the denominator zero and as we approach ne4 from the right hand side you can see in the graph it's going down towards negative infinity and as we approach it from the left hand side we're going up towards positive Infinity all right now let's talk about the hole so we see a common factor share between the numerator and the denominator and the multiplicities of that common factor x - 3 are tied they're both one that's going to produce a hole at xal 3 now to find the location of that hole what we do is we cancel out those x - 3's and we plug three into what's left over 3 + 3 is 6 3 + 4 is 7 so location of that hole is at 3 comma 67 that means the limit as we approach the hole three from the right side or the left side is going to be 67 at three there's nothing there's literally nothing it's a hole but the nothingness is located at 67 so as we approach that hole from the left or the right we get that same value of 67 so I think finding the limit as you approach a hole is really simple all you have to do is find out where the hole is located at and boom that's the answer from both directions vertical esope is the one where you have to think about all right is it approaching positive infinity or negative infinity and if you have a graph all you got to do is open your eyes and look all right let's look at example here where we don't have the function literally all we have is the graph the first thing I noticed that there is definitely a vertical ASM at positive3 so we want to figure out the behavior of our functions we approach positive3 all we got to do is open our eyes and look so the limit as we approach positive3 from the right is negative infinity and the limit as we approach positive3 from the left is positive infinity and we see that in the graph very very easily now we also see that there is a hole where is that hole located it looks like that hole is located at 2 comma 8 so as we approach two from the left and the right both sides are going to be eight because that's where the nothingness exists so as we get closer to two where nothing is we're going to get closer to that y value even though there's nothing there and that's why we're going to get eight and that's again the importance of limits it allows us to approach or allows us to think about what happens for a function as we get to somewhere not actually get there because what's happening at to DN does not exist nothing but what we could do is think about what happens as we approach that discontinuity and we see that it's going to be eight pretty simple if you ask me all right now in this next problem what we want to do is we want to think about how we could find these concepts with out a graph obviously it's much tougher but let's take our time it's overall not too bad the first we have to do is figure out what's our vertical ascope what's our whole so that way we could first make sure we're answering the right questions so the first thing of course we want to do is Factor both the numerator and the denominator and in this case we see that we have two discontinuities five and negative-1 now ne1 only turns the denominator into zero so that of course is going to be our vertical asint five turns both the numerator and the denominator into zero and it's the shared multiplicity both multiplicities of numerator denominator are one so we're going to have a whole at five all right that makes it easy so overall we have a domain of negative 1 excuse me negative Infinity to negative 1 negative 1 to 5 5 to Infinity because those two numbers have to be excluded now we notice that we do have two zeros these are real points these are not discontin these are points and that happens at 0 and four because 0 and four make only the numerator zero and they're in the domain and then we talked about that there's a vertical oope at Nega 1 and a hole at five now the first thing we want to figure out is exactly where that hole is located so what we're going to do is we're going to cover up those x - 5 factors or again let them kind of cancel out and we're going to plug five into what's left over and if you do the math real quick we get five * 9 / 6 and that's of course going to be 7.5 with that information really really easy to answer these limit questions so first let's address the whole at five because we already know it's located at 7.5 so find the limits is really really easy the limit of our function as X approaches five from the left is 7.5 as we approach five from the right it's also 7.5 now the slightly tougher question is talking about the limit as we approach the vertical esope of -1 now we know that the answers are either going to be positive infinity or negative Infinity so which one is it well to figure out the behavior as we approach negative-1 from the right pick a number on the right all we got to do is pick a number that's just to the right of neg1 like .5 all right so if we take NE .5 a value really really close to negative 1 on the right hand side in plug it in we're going to get a negative times a negative times a positive on top and then on the bottom we're going to get a negative times a another positive so that's going to result in three negatives and a positive which is going to result in an overall negative which means on the right hand side we're going down towards negative Infinity then to figure out what happens as we approach negative one from the leftand side we got to pick a number really close to negative 1 on the left hand side for example- 1.5 it's really really close and we plug in 1.5 we're plugging into all the factors we're thinking about what's happening and we're going to get a negative time a negative time a positive time a negative time another negative so that's going to be four negatives and a positive and that's going to create an overall positive so that means as we approach negative 1 from the left hand side we're going up towards positive Infinity so a little bit more thinking has to go into what happens as you approach a vertical ascope when you have a graph it's easy to just look but you just have to pick values just to the left and just to the right of that vertical ascope plug them in and determine if you're going to get a positive number or a negative number but when it comes of finding the limits towards a hole I think it's really really easy because all you got to do is find the location of that hole all right that's it for topics 1.9 and 1.10 over vertical ases and holes can be a little bit confusing so you might need a little bit more practice with identifying exactly what's happening as we approach these values and this limit notation is well kind of new to most of these students so really take your time to First process exactly what a limit means it's asking us what is be you know how the function is behaving not at a value but as we approach that value vertical ASM toes pretty simple it's going up forever or down forever if it's a hole well we're approaching the Y value of that hole which isn't too hard to find as long as we use the proper techniques all right see you in the next video hope you get tons of practice with this and I'll see you later