Episode six limits at infinity so today and next time we'll talk about limits at infinity this would be limits where X goes to infinity or X goes to negative infinity so not X approaching a number but X increasing without bound and when this happens sometimes f of X approaches a number L right so the idea here is that the limit as X goes to infinity of f of X could be equal to some number L and we could have the limit as X goes to negative infinity of f of X equaling L or these two could be infinity or negative infinity themselves I'd like to start with an example that sort of illustrates the definition of the limit a little bit better right so I hope you understand that the limit as X approaches infinity of 1 over x squared is equal to 0 we already looked at this function as X approaches zero that was positive infinity here it's sort of the other way around right we're taking X going to infinity and seeing what happens to the function 1 over x squared I hope you understand that it goes to 0 let's talk about why so what we need to do is show that 1 over x squared can be made arbitrarily close to 0 by making X sufficiently large right X is going to infinity so we have to make it very big how big that's the question so we'll let epsilon represent a small positive number and then we need to pick another number such that if X is bigger than that number it will guarantee that the absolute value of 1 over x squared minus 0 is less than an Absalon why that thing in the absolute values there that would be the distance between 1 over x squared and 0 hey that's 0 that's the L that we have here the limit it's equal to 0 right so that's what we need to show we need to show that the distance between 1 over x squared and this number 0 can be made less than every positive number so the number that we need to pick to make X bigger than that would be 1 over the square root of Epsilon let's see why that works so if we suppose that X is greater than one over the square root of epsilon then well we just square both sides x squared is going to be greater than one over epsilon and then 1 over x squared would be less than epsilon and then 1 over x squared well that that's equal to the absolute value of 1 over x squared minus 0 right that's the thing we had to show for every positive number epsilon we had to show that the distance between 1 over x squared and 0 can be made smaller than that number that's the sufficiently close to zero part by making X sufficiently big right so if epsilon is a small positive number 1 over the square root of epsilon would be a large positive number and we just make X bigger than that we'll talk more about the definition of a limit in a couple of days time but here's a brief introduction to it we won't really need to do this we just need to figure out that a function is going to a certain value as X goes to infinity or negative infinity the graphical implications of this are the horizontal asymptotes of the function if we take the limit as X approaches infinity and we get L then the horizontal line y equals L is an asymptote of f of X please make sure that your app asymptote is a line we've already talked about the vertical asymptotes the vertical asymptotes are where f of X is approaching infinity here X is approaching infinity and f of X is approaching a number so for this graph right here the limit as X goes to negative infinity we look for that horizontal line as we go off to the left that horizontal line is at y equals 2 so y equals 2 is the horizontal asymptote as we go in the negative direction as we go to the positive direction f of X goes down to this other horizontal line why it was 1 and so that limit would be 1 wife was 1 would be a horizontal asymptote this as X goes in the positive direction we do not need to have the horizontal asymptotes being the same in each direction it is most common that they are the same but they don't have to be it's also possible to have an asymptote in one direction and not have one in the other direction what can never happen is that we have two horizontal asymptotes in the same direction so you can't just have as your answer be that the horizontal asymptotes are y equals one white was two you have to specify oh it's y equals one in the positive direction and y equals two in the negative direction if you just say y equals two that means it's an asymptote in both directions so let's try to evaluate some of these without having the graph at our disposal they start with what happens for X to a power so if n is greater than zero the limit as X approaches infinity of X to the N is infinity right x squared is going to grow like a parabola right parabolas will that open up will go off to positive infinity however if n is negative the limit as X approaches infinity of X to the N would be zero this would be 1 over X to a power where that power is positive so it would be 1 divided by the previous case 1 divided by something going to infinity will go to 0 all right so X to the negative 2 power that's 1 over x squared it goes to 0 so this is where it starts you gotta understand this but I hope you already do understand this now when we have X to a power and X is going to negative infinity it becomes more complicated we really can only think about rational numbers so if n is P over Q where P and Q are both positive integers then the limit as X goes to negative infinity of X to the P over Q power is a little bit tricky if Q is even then this limit doesn't exist because we're gonna be trying to take the square root of a negative number if Q is odd and P is even then it will go to positive infinity if Q and P are both odd then this will go to negative infinity we are not in the situation where P and Q are both even we could always reduce that to a fraction where they're not both even if P over Q is negative here we're assuming P is negative and Q is positive it's a little bit easier because it will either not exist or it will go to zero we don't have to worry about it going to positive or negative infinity now if we multiply X to the N by a negative constant that might change this these really aren't that hard if you just think about them what's the limit as X goes to infinity of minus 6x cubed well X cubed as X goes to positive infinity will go to positive infinity but then I'm multiplying it by negative 6 right so this will go to negative infinity what about the limit as X goes to infinity of 1 over X to the fourth well X to the fourth is going to go to positive infinity but 1 divided by that it's going to go to 0 to fit it into the formula that I defined earlier right wait 1 over X to the fourth that's X to the negative 4 power negative 4 is negative and so this will go to 0 now what's the limit as X goes to negative infinity of X to the fifth power so we have to think about what happens to a negative number when we take it to an odd power X to the fifth when X is negative will be negative right so this will go to negative infinity let's look at the limit as X goes to infinity of 3 over the fifth root of x squared that would be 3 times X to what power think about it that's gonna be 3 times X to the negative two-fifths power negative two-fifths is negative and X is positive so we don't have any trouble there so this will go to zero right we're dividing by something that will go to infinity X to the two-fifths will go to infinity rather slowly but it will still go to infinity all right now what about the limit as X goes to negative infinity of negative two times the cube root of x I don't think about this so the cube root of x is fine when X is negative that would be a negative number it's a negative number that would go to negative infinity but I'm multiplying it by negative two so the cube root of x is negative and negative two is negative so the result will be positive and we go to positive infinity and what about the limit as X goes to negative infinity at the fourth root of x cubed so this would be X to the 3/4 power four is even so we're trying to take the fourth we're trying to take the even root of a negative number here right so this limit does not exist because the function is undefined for X being negative so now let's take a step forward in generality we'll think about polynomials polynomial will have some terms that will go to positive infinity and some terms that will go to negative infinity in the same function so how do we how do we combine these two the deposit is the one that goes to positive infinity wind doesn't lie goes to negative infinity when do they cancel each other out and we go to some finite number let's think about this alright it's pretty easy all you have to do is worry about the fastest-growing term so that's gonna be the term with the highest power so in the limit as X goes to infinity of 2x to the sixth minus 5x to the fourth minus X individually those terms alright 2x to the 6th will go to positive infinity minus 5x to the 4th will go to negative infinity minus X will go to negative infinity as well so we've got two terms going negative infinity one term going to positive infinity but the term that goes to positive infinity has the highest power so the limit as X goes to infinity of this function would be whatever the limit of the highest term is right we can actually write this limit this way write the limit as X goes to infinity of 2x to the 6th minus 5 X to the 4th minus X that limit is equal to the limit as X goes to infinity of 2x to the 6th the other two terms are irrelevant and that limit would be infinity so the limit I asked about is positive infinity all right another one how about the limit as X goes to infinity of minus X to the seventh plus three X to the fourth plus eight X cubed minus x squared plus the square root of five the only relevant term is the one with X to the seventh right so the limit as X goes to negative so the limit as X goes to infinity of minus X to the seventh X to the seventh is going to go to positive infinity but then we're multiplying it by a negative one so this will go to negative infinity and then the limit as X goes to negative infinity of 3x to the fourth minus x squared plus 10x minus nine the highest power is X to the fourth so we only have to worry about what the 3x to the fourth term goes X is negative but four is even so that means X to the fourth will go to positive infinity as X goes to negative infinity and the three is also positive so that won't change anything so this limit would be positive infinity and one more the limit as X goes to negative infinity of 2x squared minus three times the fourth root of x to the fifth power so this is not a polynomial because we have the X to the 5/4 power there let's be careful about this right the dominant term would be the 2x squared because because the squared term would grow faster than X to the 5/4 powers 2 is bigger than 5/4 but this is a place where the non-dominant terms can hold up their hand and say I have a problem right 5 X to the 5/4 power is undefined when X is negative because the 4th root is an even root so this limit does not exist because the function is undefined and then we'll look at one more class of functions for today rational functions a rational function is a polynomial divided by another polynomial so we've already looked at the limit of polynomials so all we have to really do is look at the dominant term in the top and the dominant term in the bottom for the rational function am x to the M plus some times that are irrelevant divided by B in X to the n plus some terms that are irrelevant we only have to look at the leading terms there okay so how do we do this the limit as X approaches infinity of am X to the M plus a M minus 1 X to the M minus 1 plus some other terms divided by B in X to the n plus B n minus 1 X to the N minus 1 plus some other terms is equal to the limit as X goes to infinity of the ratio of those leading terms am X to the M divided by B in X to the N the Assumption here is that neither a.m. nor BN are 0 now I can rewrite that as the limit as X goes to infinity of a M divided by B n times X to the M minus n power hey so now I just have a constant times X to a power so I just have to think about whether that power is positive or negative so if M is less than again that power is negative and so this will go to 0 if M is equal to n then we just have a constant there and that constant is a M over BN and so that limit would just be that constant if M is bigger than n this is the complicated part we know that it will be infinite because X to a positive power will go to infinity but the ratio of those coefficients am the end could be negative in which case we would go to negative infinity so we go to positive infinity if their ratio is positive we go to negative infinity if that ratio is negative and so for this limit here we have a rational function X to the fifth minus 3x to the fourth plus X cubed minus 11x squared minus X plus 1 divided by 8 X cubed minus x squared plus 15 X - we're taking the limit as X goes to positive infinity so again the only terms that are relevant are the highest power and the top and the highest on the bottom the highest power the top is 5 and that's bigger than the highest power in the bottom so the top is gonna grow faster so that means we have to think about the the ratio of those coefficients the coefficient on the top is one the coefficient in the bottom is 8 1/8 is a positive number so this limit is 1/8 times x squared as X goes to infinity so that will go to positive infinity this will eventually become very easy to use just with a little bit of practice so the limit as X goes to infinity of 2 X to the 6th power + 5 X to the 3rd power - x squared plus 7x divided by 3 minus X so the highest power in the top is 6 the highest power and the bottom is 1 so we just ignore all the other terms we have the limit as X goes to infinity of X to the 6th divided by minus X so that's going to go to negative infinity right it's basically minus 2 times X to the fifth power the X to the fifth part will go to infinity but then we multiply by negative 2 so this goes to negative infinity the limit as X goes to infinity of 375 X plus 1003 divided by x squared plus 1 again I look for the highest power in the top and highest power of the bottom highest power on the top is 1 the highest power and the bottom is 2 so we have that this is equal to the limit as X goes to infinity of 375 divided by X well X is going to grow and 375 is not so this will go to 0 it might be big initially but X is going to win X is going to grow and become bigger so the result will go to 0 eventually we're dividing 375 by a number that's a lot larger than that so we'll have a very small number and then the limit as X goes to infinity of 5 X cubed divided by 2 X cubed - X plus 13x minus 12 and so again we look for the highest power on the top and the highest power in the bottom and compare them here the highest power in the top and the bottom both of those powers are three so we look at the ratio of their coefficients and that's going to be five halves so this function has a horizontal asymptote of y equals five halves in the positive direction so for the next one let's look at the same function but instead of going to positive infinity we'll go to negative infinity we don't really do anything different here we look for this the highest we look for the highest power in the top and we look for the highest power in the bottom and again we see that they are the same and so this would be the ratio of their coefficients so this one would also be five halves so y equals five halves is horizontal asymptote in the negative direction as well and one more the limit as X goes to negative infinity of five x squared minus 17 divided by 3x plus 4 again we look for the highest power in the top the highest power in the bottom here we see that the highest power in the top is bigger so it will be infinite we just have to figure out if it's positive infinity or negative infinity well 5 x squared is going to go to positive infinity because 2 is even but 3 X will go to negative infinity so the higher power of the top tells us the infinity part but here the bottom is going to make this go to negative infinity so this limit is negative infinity with these rational functions it will get pretty easy with some practice right you just if the highest power in the bottom is bigger you go to 0 and you have a lot horizontal asymptote of y equals 0 if the highest power on the top and the highest power in the bottom of the same you go to whatever the ratio of their coefficients is so that would be your horizontal asymptote and then if the highest power in the top is bigger than the one in the bottom you'll go to either positive infinity or negative infinity you'll have to be careful about that how to think each one individually about whether that is positive or negative and you would not have horizontal asymptotes in that situation