End behavior of rational functions

so we're given this function f of X and it equals this rational expression over here and we're asked what does f of X approach as X approaches negative infinity so as X becomes more and more and more and more negative what does f of X approach and like always pause the video and see if you can think about that on your own well one thing that I like to do when I'm trying to consider the behavior of a function as X gets really positive or really negative is to rewrite it so f of X I'm just rewriting it once is equal to 7x squared minus 2x over 15 X minus 5 now an interesting technique to think about what happens to the different terms as X gets very positive or X gets very negative is to divide both the numerator and the denominator by the highest degree term of X in the denominator and the highest degree term of X in the denominator is the first degree term we just have we have just a single X there so let's multiply both the numerator and the denominator by 1 over X or another way of thinking about is we're dividing both the numerator and the denominator by X and if we're doing the same thing to the numerator and denominator if we're multiplying or dividing them by the same value I should say well then I'm just really just multiplying it by one so I'm not changing its value but this will make it a little bit more interesting a little bit easier for us to think about what happens when X becomes very very very negative so 7x squared divided by X or being multiplied by 1 over X is going to be 7x 2x times 1 over X or 2x divided by X is just 2 and then all of that over 15 X divided by X or 15 X over X is just going to be 15 and then you have 5 over X 5 times 1 over X is equal to 5 over X minus 5 over X now this is this is equivalent for our purposes to what we started with but it makes a little bit easier to think about what happens when X gets very very very very negative well when X gets very very very very very very negative this is going to become a very large negative number you subtract two from it really won't matter much you divide that by 15 well that's not going to matter much and this is just going to become very very very small you're taking five and you're dividing it by ever larger negative numbers or more and more negative numbers so this right over here is going to go to zero this thing over here is going to go towards infinity and if you or I should say it's going to go towards negative infinity seven times you know for the seven times a negative trillion seven times a negative Google seven times a negative Googleplex we're getting more and more negative numbers this is going to get this is going to approach negative infinity it doesn't matter that you're subtracting two from that in fact that will get even more negative and it doesn't matter if you then divide that by 15 you're still approaching negative infinity if you had negative if you had a arbitrarily negative number you divide by 15 you still have an arbitrarily negative number and so you could say that this is going to go to negative infinity now another way that you could have thought about it and this is actually how I do think about it when I'm trying to when when I'm when I see these types of problems I say well what are which-which terms in the numerator in the denominator are going to dominate and what do I mean by dominate well as X gets very positive or X gets very negative another way to think about is the magnitude of X gets large the absolute value of x gets large the second the higher degree terms are going to grow much faster than the lesser degree terms and so we could say that for for large X for large for large X and when I say outlook large I mean high absolute value high absolute value and if we're going to negative infinity that's high absolute value so f of X is going to be approximately equal to the highest degree term on the top which is 7x squared divided by the highest degree term on the bottom 15 X is going to grow in fact this is right over here is constant so as this becomes larger and larger and larger this is going to matter a lot lot less so it's going to be approximately that which is equal to 7x over 15 well even here think about what happens when X becomes very very negative here well you're just going to get larger you're going to get more and more and more negative values for f of X so once again f of X itself is going to approach is going to go to negative infinity as X goes to negative infinity let's do another one of these so here they're telling us to find the horizontal asymptote of Q and you could find a horizontal asymptote you could think about it as what is the function approaching as X becomes as X approaches infinity or as X approaches negative infinity and just as a couple of examples here it's not necessarily the one I'm Q of X that we're focused on but you could imagine a function let's say it has a horizontal asymptote at at Y is equal to two so that's y is equal to two there let me draw that line so let's say it has a horizontal asymptote like that well then the graph could look something like this it could look it could let me draw a couple of them that have horizontal asymptotes so maybe it's over here it does some stuff but as X gets really large it starts approaching it the function starts approaching that y equals two without ever quite getting there and it could do that on this side as well as X becomes more and more as X becomes more and more negative it's as more as it gets more negative approaches it without ever getting there or it could do something like this you could have if it has a vertical asymptote - it could look something like this where it approaches it approaches the horizontal asymptote from below as X becomes more negative and from above and from above as X becomes more positive or vice versa or vice versa so this is just a sense of what a horizontal asymptote is it's really what's the behavior what value is this function approaching as X becomes really positive or X becomes really negative well let's just think about we could essentially do what we just did in that last example what happens if we were to if we were to divide all of these terms by the highest degree term the denominator well if we divide so Q of X is going to be equal to the highest degree term for the denominator is X to the 9th power so we could say 6 6 X to the 5th divided by X to the 9th is going to be 6 over X to the 4th and then minus 2 times X to the 9th all of that over 3 over I'm going to divide this by X to the 9th X to the seventh plus 1 well if X approaches positive or negative infinity 6 divided by arbitrarily large numbers that's going to go to 0 2 divided by arbitrarily large numbers whether they are positive or negative that's going to go 0 go to 0 so your numerator is clearly going to go to 0 this term of the denominator 3 divided by arbitrarily large numbers whether those large whether we're going in the positive or the negative direction it is going to get is going to approach 0 it'll approach 0 from the negative direction if we're approaching if our or we could say from below if if we're dealing with very negative X's and we're dealing with very positive X's then we're going to approach 0 from above we're going to get smaller and smaller positive values so all of these things go to 0 and this right over here is going to be would stay at 1 and so if you're approaching 0 in your numerator and approaching 1 in your denominator the whole thing is going to approach 0 so in the case of Q of X you have a horizontal asymptote at Y is equal to 0 I don't know exactly what the graph looks like but we could draw a horizontal line at y equals 0 and it would approach it it would approach it from above or below let's do one more what does f of X approach as X approaches negative infinity well let's divide all of these let's divide all of these terms by the highest degree that we see in the denominator we see an X to the 4th so 3x to the 4th divided by X to the 4th is 3 minus 7 over x squared I'm just dividing by X to the 4th minus 1 over X to the 4th over X to the 4th divided by X to the 4th is 1 - 2 over X + 3 X to the fourth this is an equivalent this right over here is for our purposes for thinking about what's happening on a kind of an end behavior as X approaches negative infinity this will do I've just divided everything by X to the fourth and so what's going to happen as X approaches negative infinity this is going to approach zero this is going to approach zero this is going to approach zero and this is going to approach zero and so it's all that stuff approaches zero what we're left with is we're going to approach we're going to approach three over one or we could say just three another way you could think about doing these is look at the highest degree terms 3x to the fourth X to the fourth ignore everything else because they're going to be overwhelmed by these higher degree terms so you could say f of X is approximately equal to 3x to the fourth over X to the fourth for large large magnitude X magnitude X and very negative is still a very large magnitude large absolute value and so 3x to the fourth divided by X to the fourth f of X is going to be approximately equal to three what's going to approach three so that's another way that you could think about it