Limits at infinity of quotients (Part 2)

let's do a few more examples of finding the limit of functions as X approaches infinity or negative infinity so here I have this crazy function 9 X to the seventh minus 17 X to the 6 plus 15 square roots of X all of that over 3x to the seventh plus 1000 X to the fifth minus log base two of X so what's going to happen as X approaches infinity and the key here like we've seen in other examples is just to realize which terms will dominate so for example in the numerator out of these three terms the 9 X to the seventh is going to grow much faster than any of these other terms so this is the dump do not this is the dominating term in the numerator and the denominator 3x to the seventh it's going to grow much faster than an X to the fifth term I'm not definitely much faster than a log base 2 term so at infinity as we get closer and closer infinity this function is going to be rough it's going to be roughly equal to 9 X to the seventh over 3x to the seventh and so we can say especially since as we get larger and larger as we get closer and closer to infinity these two things are going to get closer and closer to each other we can say this limit is going to be the same thing as this limit which is going to be equal to the limit as X approaches infinity well we can just cancel out the X to the 7 so it's going to be 9 thirds or just 3 which is just going to be 3 so that is our limit as X approaches infinity of all of this craziness now let's do the same with this function over here once again crazy function we're going to negative infinity but the same principles apply which terms which terms dominate as the absolute value of x get larger and larger and larger as X gets larger in magnitude well in the numerator it's the 3x to the third term in the denominator it's the 6 X to the fourth term so this is going to be the same thing as the limit of 3 X to the third over 6 X to the fourth as X approaches negative infinity and if we simplify this this is going to be equal to the limit as X approaches negative infinity of 1 over 2 X and what's this going to be well if the denominator even though it's becoming a larger and larger and larger negative number it becomes one Oh or a very very large negative number which is going to get us pretty darn close to zero just as one over X as X approaches negative infinity gets us close to zero so this right over here the horizontal asymptote in this case is y is equal to zero and I encourage you to graph it or try it out with numbers to verify that for yourself the key realization here is to simplify the problem by just thinking about which terms are going to dot which terms are going to dominate the rest now let's think about this one what is the limit of this crazy function as X approaches infinity well once again what are the dominating terms in the numerator it's 4 X to the fourth and the denominator is 250 X to the third these are the highest degree terms so this is going to be the same thing as the limit as X approaches infinity of 4 X to the fourth over 200 over 250 250 X to the third which is going to be the same thing as the limit of let's see 4 well I could just read this is going to be the same thing as we could divide 200 in well I'll just leave it like this it's going to be the limit of 4 over 250 X to the fourth divided by X 2/3 just x times X as X approaches infinity or we could even say this is going to be for 250 it's for 250 it's times the limit as X approaches infinity of X now what's this what's the limit of X as X approaches infinity what is just going to keep growing what forever so this is just going to be this right over here is just going to be infinity infinity times some number right over here is going to be infinity so the limit as X approaches infinity of all of this it's actually unbounded its infinity and a kind of obvious way of seeing that right from the get-go is to realize that the numerator is has a fourth degree term while the highest degree term in the denominator is only a 3rd degree term so the numerator is going to grow far faster than the denominator so the numerator is growing far faster than far faster than the denominator you're going to approach infinity in this case if the numerator is going far if the numerator is going far slower than the denominator if the denominator is going far faster than the numerator like this case you're then approaching zero so hopefully find that a little bit useful