Functions with same limit at infinity

the goal of this video is to get an appreciation that you could have many in fact you could have an infinite number of functions that have the same limit as X approaches infinity so if we were to make the general statement that the limit of some function f of X as X approaches infinity is equal to 3 what I want to do in this video is show some examples of that and to show that we can keep creating more and more examples really an infinite number of examples where that is going to be true so for example we could look at this graph over here and you in other videos will think about why this is the case but just think about what happens when you have very very large X's when you have very very large X's the +5 doesn't matter as much and so it gets closer and closer to 3 x squared over x squared which is equal to 3 and you could see that right over here it's graphed in this green color and you can see even when X is equal to 10 we're getting awfully close to 3 right over there let me zoom out a little bit so you see our axes so that is 3 let me draw a dotted line at the asymptotes that is y is equal to 3 and so you see the function is getting closer and closer as X approaches infinity but that's not the only function that could do that as I keep saying there's an infinite number of functions that could do that you could have this somewhat wild function that involves natural logs that too well as X approaches infinity it is getting closer and closer to 3 it might be getting closer to 3 at a slightly slower rate than the one in green but we're talking about infinity as X approaches infinity this thing is approaching 3 and as we've talked about in other videos you could even have things that keep oscillating around the asymptotes as long as they're getting are as long as it getting closer and closer and closer to it what as X gets larger and larger and larger so for example that function right over there let me zoom in so let's zoom in when let's say when X is equal to 14 we can see that they're all approaching 3 the purple one is oscillating around it the other two are approaching 3 from below but as we get much larger let me actually zoom out of ways and then I'll zoom in so let's get 2 really large values so actually even 100 isn't even that large if we're thinking about infinity even a trillion wouldn't be that large everything about it 50 but let's go to 200 200 is much larger the numbers we've been looking at it let me zoom in when X is equal to 200 and you can see we def we have to zoom in an awfully lot an awful lot just to even see that the graphs still aren't quite stabilized around the asymptotes but they are a little bit different than the asymptotes I am really zoomed in let me look at the scale this is each of these are now a hundred each square and so we've gotten much much much closer to the asymptotes back the green function we still can't tell the difference you can see the calculation this is up to three or four decimal places we're getting awfully close to three now but we aren't there so the green functions got there the fastest is an argument but the whole point of this is to emphasize the fact that there's an infinite number of functions for which you could make the statement that we made that the limit of the function as X approaches infinity in this case we said that limit is going to be equal to three and I just picked three arbitrarily this could be true for any or any function I'm trying to I didn't realize how much I had loomed in so let me now go back to the origin where we had our original expression so there we have it I maybe I could zoom in this way so there you have it limit of any of these as X approaches infinity is equal to three