Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College we say that f(x) has the limit L as x approaches infinity and write limit of f(x) as x goes to infinity equals L if for every number epsilon greater than zero, 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(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 one over x, what happens when x gets really, really close to zero 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 gonna go out to infinity if we look at limit as x goes to zero 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(x) if either the limit as x approaches some constant from the right side of f(x) equal positive and negative infinity or the limit as x approaches a from the left side of f(x) equal positive and negative infinity horizontal asymptotes a line y equal b is a horizontal asymptotes of the graph of a function y equal f(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(x) equal b let's look at some examples if we have the function of one divided by x minus three and we wanna know what happens when the limit of x as x approaches three from the right first of all when we look at this, we're gonna realize that there's a vertical asymptote every time the denominator equals zero so we're gonna have a vertical asymptote at x equal three we're goona have a horizontal asymptotes, 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 gonna be really really small sowe're gonna have a horizontal asymptote of y equal zero the easiest way to do that is if the degree in the bottom is bigger the horizontal asymptote's always zero if the degree in the numerator and the denominator are the same it's the leading coefficients and if the degree in the numerator's bigger we have to do long division and look for an oblique asymptote so this is gonna be enough information to get us started. If we have a vertical asymptote at x equals three and a horizontal at y equal zero if we thought about putting in x equal zero to give us a point to get started. If we put in x of zero we know that f(x) equal one over x minus three so f of zero is one over a zero minus three or negative one-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 gotta be close to the asymptote at negative infinity. We've gotta go through that point and we've gotta get close to the other asymptote the vertical at x equal three x equal three 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 gotta 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(4). We have one over four minus three one over one, one which would give us a positive value so our question actually said what happens to the limit as x approaches three 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 y's doing? The y's are going out to infinity. So our answer to this first one would be infinity looking at this next example we're gonna start with our vertical asymptote is going to occur at x equal negative five because the denominator can never be zero our horizontal asymptote the degree here is the same on top and bottom so the horizontal is gonna be the leading coefficients or in this case 3/2 x-intercepts basically we set y equal to zero to find an x-intercept and what's gonna happen every time is the denominator is gonna cancel so we could really think of this as just setting the numerator equal to zero, because zero times the denominator is always gonna be zero so we're gonna get an x intercept at (0,0) y-intercept we get by putting zero in for x and in this case it's also going to be zero, so the point (0,0) so when we graph this, we get x equal negative five we get y equal 3/2 or one and a half and we know the point (0,0) is the only intercept so when we're at x going out to infinity, we've gotta be close to the horizontal at this point (0,0) that actually came from the multiplicity in this numerator and it's multiplicity is one so if the y-values on one side are positive, the y-values on the other side are gonna be negative and it's gotta get close to the vertical asymptote now the vertical asymptote's degree came from this denominator and its degree is one so if the y-values on one side are negative the y-values on the other side are gonna have to be positive we've gotta 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 negative five what happens to x as we go to negative five from the left hand side? So if we get closer and closer and closer to negative five from the left hand side, we're going out to infinity looking at some more examples here we're gonna have vertical asymptotes at x equal zero and also at x equal negative one horizontal asymptote at y equals zero 'cause 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 equals zero, x equal negative one, y equals zero if we thought about looking at a positive value let's say f of one we'd get negative over one squared one plus one so we would get negative one over two 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 equal zero's gonna be an arc like this in between zero and negative one we can look at multiplicities. The multiplicity for this x equal zero came from this term here and the multiplicity is two, so it's even hence the y-values on one side have gotta 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 graphs so we're just gonna go up and come back down to the next asymptote the x equal negative one asymptote comes from this term which has an 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 gonna 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 zero 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 zero of one over x to the one third squared if we think about it as this, we know that whatever is on the inside is gonna get squared and hence it's gonna be positive so we know we're gonna have a vertical asymptote at x equals zero we're gonna have a horizontal asymptote at y equal zero so when we look at this graph if we thought about f of one let's say to get a starting point we have one over one to the two-thirds or one so we know we've gotta get close to the asymptotes. This square, the multiplicity is even so the y-values on one side of that vertical asymptote have gotta 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 zero from both the left and the right we see that this is gonna go out to positive infinity if we look at the limit is x approaches negative pi 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 gonna have a graph that looks like this where the full period is two pi half the period's pi asymptote at pi halves and three pi halves. The asymptotes have to occur when cosine was zero which is at pi halves and three pi halves it repeats so we also have one at negative pi halves the pattern keeps going negative three pie halves when we look at this graph we wanna know what happens at negative pi halves as x goes to negative pi halves from the right hand side so when we look at what happens as x gets closer and closer and closer to negative pi halves from the right hand side we'd have infinity if this problem had said what's the limit as x approaches negative pi halves from the left hand side from the left hand side we get closer and closer and closer to that negative pi halves and we'd get negative infinity if it had just said what's the limit as x goes to negative pi 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 one's negative infinity looking another example we could graph x divided by x squared minus one. By factoring that denominator, we'd have a vertical asymptote at x equal one and also x equal negative one horizontal asymptote would be at y equals zero because the degree and bottom is bigger. We'd have an x-intercept at the point (0,0) and we'd have a y-intercapt at (0,0) so if we make ourselves a rough sketch of this one, x equal one, x equal negative one, y equal zero 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 wanna put in a value let's go f of two maybe, we'd get two over two squared minus one so that would give us a positive value that one starting value will help me figure out how the rest of the graph's gonna look. At x equal one 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 or negative we're gonna come up to our x-intercept our x-intercept comes from here and it's 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 okay to go through or cross or touch even a vertic- or a horizontal asymptote or an oblique our last one our x equal negative one comes from that term which also has an odd multiplicity so if the y's are positive on one side we know the y's 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 inside we have our y-values going to negative infinity if we look at negative one from the right hand side we're gonna see that it's going off to infinity and if we look at negative one from the left side we have negative infinity our last example is going to being an oblique and so if we have y equal x squared plus one over x minus one and we wanna figure out the limit as x goes to one from the right hand side of f(x) instead of y, let's call that f(x) what happens to the limit as x goes to one from the left hand side so here that 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 times x, I'm gonna put a zero x as a placeholder because I encourage students to keep the x's all inline so x times x minus one is x squared minus x we subtract bring down the next term plus one we subtract and we get a remainder of two over x minus one now what happens is when we get way out to infinity or negative infinity, this two over x minus one really goes to zero so our oblique, when we're way out in infinity and negative infinity is y equals x plus one we have vertical asymptotes still where the denominator is undefined so we have x equal one and we have y equal x plus one we have when we think about x-intercepts we don't have any because zero doesn't ever equal x squared plus one on real numbers y-intercept, we're gonna have (0,-1) so that one point is gonna be enough to tell me how this graph's gonna 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 were going to infinity if we're looking at one from the left hand side we're going to negative infinity