Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. We say that F of X has the limit L as X approaches Infinity and write limit of F of X as X goes to Infinity equals L if for every number epsilon greater than 0 there exists a corresponding number M such that for all X where X is greater than M implies that the absolute value of the difference between the function and the limit is less than that epsilon or the error value. We also have the limit as X approaches negative Infinity of F of X equaling L when X is smaller than some N implies that the absolute value of the difference between the function and the limit is less than the error. What that really says is that infinite limits given any positive real number B, the value of F becomes larger still. So if we look at an example of the function Y 1 / X, what happens when X gets really, really close to 0 from the right hand side? So what we're looking at is we're looking at coming in to this function to zero from the right hand side. And we can say that the Y values keep getting bigger and bigger and bigger. So it's going to go out to Infinity. If we look at limit as X goes to 0 from the left side, so now we're coming in from the left side, what's happening to our Y values? Our Y values are going out to negative Infinity, so we want to know about vertical asymptotes. A line X equal a is a vertical asymptote of the graph of a function Y equal F of X if either the limit as X approaches some constant from the right side of F of X equal positive or negative Infinity, or the limit as X approaches a from the left side of F of X equal positive or negative Infinity. Horizontal asymptotes. A line Y equal B is a horizontal asymptote of the graph of a function y = F of X if either as X the limit as X approaches Infinity. So as X gets really big of the function equals some constant B or the limit as X goes to negative Infinity of F of X equal B. Let's look at some examples. If we have the function of one divided by X -, 3, and we want to know what happens when the limit of X as X approaches 3 from the right. First of all, when we look at this, we're going to realize that there's a vertical asymptote every time the denominator equals 0. So we're going to have a vertical asymptote at X = 3. We're going to have a horizontal asymptote. If we think about having the denominator get bigger and bigger and bigger as we go out to Infinity, well, what's one over something really, really big? One over something really, really big is going to be really, really small. So we're going to have a horizontal asymptote of y = 0. The easiest way to do that is if the degree in the bottom is bigger, the horizontal asymptotes always 0. If the degree in the numerator and the denominator are the same, it's the leading coefficients, and if the degree in the numerator is bigger, we have to do long division and look for an oblique asymptote. So this is going to be enough information to get us started if we have a vertical asymptote at X equal 3 and a horizontal at Y equal 0. If we thought about putting in X = 0 to give us a point to get started. If we put in X as zero, we know that F of X equal 1 / X - 3, so F of 0 is 1 / 0 - 3 or -1 third. So that one point there is enough to tell us how the rest of the graph should look. Knowing our asymptotes, we know we've got to be close to the asymptote at negative Infinity. We've got to go through that point and we've got to get close to the other asymptote. The vertical at X equal 3, X equal 3 has a degree or a multiplicity of one. It was to the first power. So if the Y values are negative on one side, they've got to do the opposite or be positive on the other. If you don't see that, you can always put in a value, let's say F of four. We have 1 / 4 -, 3, one over 11, which would give us a positive value. So our question actually said, what happens to the limit as X approaches 3 from the right hand side? So as we're getting closer and closer and closer to three from the right hand side, what are the YS doing? The YS are going out to Infinity. So our answer to this first one would be Infinity. Looking at this next example, we're going to start with our vertical asymptote is going to occur at X equal -5 because the denominator can never be 0, our horizontal asymptote. The degree here is the same on top and bottom, so the horizontal is going to be the leading coefficients, or in this case 3 halves X intercepts. Basically we set Y equal to 0 to find an X intercept, and what's going to happen every time is the denominator is going to cancel. So we could really think of this as just setting the numerator equal to 0 because zero times the denominator is always going to be 0. So we're going to get an X intercept at 00 Y intercept we get by putting 0 in for X, and in this case it's also going to be 0, so the .00. So when we graph this, we get X equal -5 we get Y equal 3 halves or 1 1/2. And we know the .00 is the only intercept. So when we're at X going out to Infinity, we've got to be close to the horizontal at this point 00. That actually came from the multiplicity in this numerator, and its multiplicity is 1. So if the Y values on one side are positive, the Y values on the other side are going to be negative, and it's got to get close to the vertical asymptote. Now, the vertical asymptotes degree came from this denominator, and its degree is 1. So if the Y values on one side are negative, the Y values on the other side are going to have to be positive, and we've got to get close to the asymptotes. We also realize that we don't have any intercepts out this way, so we couldn't cross this graph. So what happens to -5? What happens to X as we go to -5 from the left hand side? So if we get closer and closer and closer to -5 from the left hand side, we're going out to Infinity. Looking at some more examples here, we're going to have vertical asymptotes at X = 0 and also at X equal -1 horizontal asymptote at y = 0. Because the degree on bottom is bigger, we don't have any X intercepts or Y intercepts. So if we put all this on our graph, X = 0, X equal negative one, y = 0. If we thought about looking at a positive value, let's say F of one, we'd get -1 / 1 ^2 1 + 1, so we would get -1 / 2. So at one we know a point down here, we know we have to get close to the asymptotes. So we know that to the right of X = 0 is going to be an arc like this. In between 0 and -1, we can look at multiplicities. The multiplicity for this X = 0 came from this term here, and the multiplicity is 2, so it's even. Hence, the Y values on one side have got to be the same as the Y values on the other. Now we don't have any X intercepts, so we know we can't cross the graph. So we're just going to go up and come back down to the next asymptote. The X equal -1 asymptote comes from this term, which has a odd multiplicity. It's to the first power. So if the Y values are negative on one side and it's an odd multiplicity, they're going to be the opposite on the other. So that's a rough sketch of what the graph would look like. And what it wants to know is what happens when X goes to 0. Now we have to pay attention this time because it didn't say from left to right. So we have to look if we're coming in from the both the left and the right, it has to go to the same value for the limit to exist. And in this case they do. It goes to negative Infinity. Looking at another one, this one we could think of as the limit as X goes to 0 of 1 / X to the 1/3 ^2. If we think about it as this, we know that whatever's on the inside is going to get squared, and hence it's going to be positive. So we know we're going to have a vertical asymptote at X = 0. We're going to have a horizontal asymptote at y = 0. So when we look at this graph, if we thought about F of one, let's say to get a starting point, we'd have 1 / 1 to the 2/3 or 1. So we know we've got to get close to the asymptotes, the square, the multiplicity is even. So the Y values on one side of that vertical asymptote have got to be the same type of values, so positive on one side, positive on the other. When we look at what happens when X goes to 0 from both the left and the right, we see that this is going to go out to positive Infinity. If we look at the limit as X approaches negative π halves from the right side for secant X, our graph of secant X back from our trigonometry days, secant X is really just one divided by cosine. So we know we're going to have a graph that looks like this, where the full. Is 2π half. The periods π asymptote at Pi halves and at three Pi halves. The asymptotes have to occur when cosine was 0, which is at π halves and three Pi halves. It repeats so we also have one at negative π halves. The pattern keeps going -3 Pi halves. When we look at this graph, we want to know what happens at negative π halves as X goes to negative π halves from the right hand side. So when we look at what happens as X gets closer and closer and closer to negative π halves from the right hand side, we'd have Infinity. If this problem had said what's the limit? As X approaches negative π halves from the left hand side. From the left hand side, we'd get closer and closer and closer to that negative π halves and we get negative Infinity. If it had just said what's the limit as X goes to negative π halves of secant X, it was. It does not exist because the left side and the right side would not be the same values one's Infinity, 1's negative Infinity. Looking at another example, we could graph X / X ^2 -, 1 by factoring that denominator. We'd have a vertical asymptote at X equal 1 and also at X equal -1. Horizontal asymptote would be at y = 0. Because the degree on bottom is bigger, we'd have an X intercept at the .00 and we'd have AY intercept at 00. So if we make ourselves a rough sketch of this One X equal 1, X equal negative one y = 0. Now we can always cross a horizontal and an oblique asymptote. We can never cross a vertical asymptote. The reason is a horizontal or an oblique is actually only true at Infinity and negative Infinity. So here, if I want to put in a value, let's go F of two, maybe we'd get 2 / 2 ^2 -, 1, so that would give us a positive value. That one starting value will help me figure out how the rest of the graph is going to look at X equal 1 came from this term here this line. It's an odd multiplicity. So if the Y values on one side are positive, the Y values on the other have to be the opposite, are negative. We're going to come up to our X intercept. Our X intercept comes from here, and its multiplicity is odd, so the Y values on one side are negative, Hence the Y values on the other side are positive. Remember, it's OK to go through or cross or touch even, or a horizontal asymptote or an oblique. Our last one, our X equal -1, comes from that term, which also has an odd multiplicity. So if the YS are positive on one side, we know the YS have to be negative on the other. Now we can put together several questions from the one graph. What happens to X as we approach one from the right hand side? Well, if we're approaching one from the right hand side, our Y values are going off to Infinity. If we approach one from the left hand side, we have our Y values going to negative Infinity. If we look at -1 from the right hand side, we're going to see that it's going off to Infinity. And if we look at -1 from the left side, we have negative Infinity. Our last example is going to be an oblique and so if we have Y equal X ^2 + 1 / X -, 1, and we want to figure out the limit as X goes to one from the right hand side of F of X instead of Y, let's call that F of X. And what happens to the limit as X goes to one from the left hand side? So here the degree on top is bigger. So we actually have to do polynomial division, or in this case we could do synthetic division. We know that X * X. I'm going to put in a 0X as a placeholder because I encourage students to keep the X's all in line. So X * X - 1 is X ^2 -, X we subtract, bring down the next term plus one, we subtract and we get a remainder of 2 / X -, 1. Now what happens is when we get way out to Infinity or negative Infinity, this 2 / X - 1 really goes to 0. So our oblique when we're way out in Infinity and negative Infinity is Y equal X + 1. We have vertical asymptotes still where the denominator is undefined. So we have X equal 1 and we have Y equal X + 1. We have. When we think about X intercepts, we don't have any because 0 doesn't ever equal X ^2 + 1. On real numbers. Y intercept we're going to have zero -1. So that one point is going to be enough to tell me how this graph is going to go. I can't cross the X axis, and I know I have to be close to the intercepts or not. The intercepts, the asymptotes. Sorry, this vertical asymptote came from here. It's an odd multiplicity. So if the Y values on one side are negative, the Y values on the other side have to be positive. So if we're looking at one from the right hand side, we're going to Infinity. If we're looking at one from the left hand side, we're going to negative Infinity.