Now we're comparing a bunch of functions to X square to see how they grow with respect to X squared. So here we have X squared plus four at now, we can expand this out and that bombs becomes one plus four over X. And as X goes infinity um this goes to one. Um So that means that because this is a constant, that means these two functions grow at the same rate which you kind of expect because you know as X gets large this term is going to be negligible compared to this. And so they both look like growing as as quadratic functions as X gets large. Now we have X to the fifth minus X squared. Um expanding that out, we get x cubed minus one and as X goes infinity that clearly goes to infinity. So this X X to the fifth minus X squared although um grows much faster than X squared. And again we could see that because the highest power here is greater than the highest power here. Um Now we've got a square root, so we have the square root of X squared or X to the fourth plus X squared. Now again we can kind of look at this and say, ok as X gets very large, that's the fourth is gonna get very very big compared to X square, so that's going to be negligible. So this is gonna look like X squared. And so again, we would expect these two things to grow at the same rate. So let's check that out. So we can expand, pull this inside the square root and then pull the limit inside the square root, this becomes one plus one over X squared. The limit of that is one. The square root of that is one. And so indeed they do grill at the same rate. Um Now we have X plus three all squared and again we should see that we'd expect um you know this is a quadratic function and so is this, so if we expand this out and then divide through by this, we get one plus six over X plus nine over X squared. And taking the limit as x goes infinity, this goes to one. And so indeed these to do grow at the same rate. Now we're looking at X. Time for natural log of X. Okay, so this is an interesting function because um in fact we would expect this because natural lagerback's grow slower than X, we would expect this to grow slower than this. So let's see if that's the case. So we take the limit um we can see that in X cancels out. So we get the limit of natural log of X over X. Um Using logic college rule we get that this is the limit of one over X, which indeed goes to zero. And so X X times the natural log of X indeed grow slower than X square. Um Now we have either the X. We want to check that compared to X square and we would expect this to grow faster than X squared because you know either anything to the power of X or anything greater than one to the power of X grows grows very large very quickly. Um So we can use capitals rule, take the derivative appear and we just get this natural log of two term here and here we get to X. That because rule again get to to the X. Times again the natural log of two squared and then this just nominated becomes too and then this clearly goes to infinity. So clearly either the X grows faster than X squared. Mhm. Now this is a little interesting function because now we have something that grows faster than X squared times, something that grows slower the next word. And in fact this grows Um this goes to zero as x goes to infinity. So what we can see here is that if we divide that by X X cubed uh X squared we with the limit of X times E to the minus X. And that's just the limit of X over E. To the X. Capitals rule says that that's the limit of one overeating the X. And that goes to zero. So in fact this grows slower than this because basically in fact the limit of just the limit of um X cubed either the X that limited X cubed times either minus X actually goes to zero also. So Um this whole numerator here goes to zero as x ghost infinity, where this goes to infinity. Now we have um eight X squared, so they left us with an easy one here. Um X squared cancel out. So we get the limit as X scores infinity of eight, which is just eight, so eight X squared grows is the same rate as X squared.