### Stony Brook MAT 125 Spring 2015 Video Introduction to Limits, pre-lecture

 Start The topic for today is limits and this is a critical piece for understanding calculus calculus is built on the idea of limits well actually they develop calculus before they know how to do limits but that they were a lot of mistakes so in order to make everything work right the idea of limits was introduced to make sure that there was everything was on a solid foundation now one of our motivating examples for calculus is the 0:30 idea of position or distance traveled vs velocity and so if you know for example that you go on a car trip and you travel 100 miles in two hours then we get an average of 50 miles per hour over that will period but perhaps in the first part you 1:02 want a little slow and you went 20 miles in the first hour and the second hour you manage to get on the expressway and you went 80 miles in the second actually maybe we did it the other way around you want eight miles in the first hour and then you got pulled over and you got a ticket you were stopped trouble there and then the second hour 1:34 you only went 20 miles because you spent a lot of time sitting by the side of the road while the guy was writing you a ticket so here we have a very difference in velocity and we averaged over smaller times we get different rates say if we look more closely at what you were doing during this middle . you were in fact going 0 miles for half an hour and so your average speed is zero and we look on 2:08 smaller and smaller and smaller range as we get some time is very different very different speeds if we look at this in terms of a graph in the first hour you didn't go anywhere at all but you went there quite then you started going quite quickly then you got pulled over and you didn't go anywhere for a while and then you sped up and you 2:33 went somewhere now a question we might ask is how fast were you going right at the moment when you got pulled over and the way that we would have to figure that out or is by averaging the distance over smaller and smaller intervals or zooming in on this graph and looking very closely and what's going on and then we compute the slope of the tangent line but in order to do that computation 3:00 we may have to divide by a very small time interval and we want to know what happens as we let that time interval go smaller and smaller and smaller so instead of looking at this distance versus velocity problem what we can do is look at just the general question of suppose that I have some function that I 3:32 can't compute at some specific moment but i can compute values nearby in this case this would be I can look at how fast you go how far you travel in any finite time interval whether that's one hour 1 minute 1 second one tenth of a second but i want to know what happens as we get very close so instead of taking the situation of speed vs distance i'm going to just take a general 4:04 function so for example suppose we look at some function like f of X is sine 3x divided by X this is a perfectly good function except if X is not00 this problem this function has a problem at zero because certainly absolute 0 is 0 / 0 which is undefined 4:40 we don't know what number we can assign to that but if we look at this function for values very close to zero so for example if we look at f of x computed some if we look at f of 0.2 we get a perfectly good number of 2.8 23 5:03 look at f of 0.1 we get another nice number of f of 0 (can't read my notes) f of 2.9552 f of 0.01 is 2.99 2.9887 and so and if we look at smaller and 5:39 smaller numbers we get lots of 9's here it looks like the answer is getting very close to three similarly if we look at negative numbers to look negative numbers and s0 0.2 its the same as that because the sine of negative 0.6 is- whatever it is we're 6:05 dividing a negative x negative we get the same answer so this is exactly the same if we look on either side of zero and if we draw the graph this function will get something that looks roughly 6:32 like this this is PI negative PI and so on so we'll get a nice function and it sure looks like we want this value to be three that doesn't mean that f of 0 is 3 what it means is as we look closer and closer and closer but not exactly at 0 the value gets closer and closer and closer to three which is what we see 7:04 here by plugging in so we would say in this case --- this is not a proof, this is just a justification. To actually prove that sin(3 x)/x is actually 3 is a little challenging, we will do it but not yet --- but we would say in this case if we could justify this completely; I can we do it later --- is that the limit as X goes to 0 of 7:33 sin(3x) divided x is in fact three. In this notation each of these little symbols mean something here we write limit to mean that we're taking the limit and this X arrow 0 means that X is closer to closer and closer to 0 but it never is 0 so we have this in general if 8:06 we write something like the limit as whatever variable we choose let's call it Y now and it goes to some number like 5 of some function let's call an H and we say it's some number like 10 so suppose we have this statement limit y goes to 8:30 five h y is 10 this means the words as y get's closer and closer to five hy gets closer and closer to10 it doesn't mean that h of x or h of y is 9:11 10 we haven't defined h5 we don't care what the value is but it doesn't mean that it gets really close we can say that in a very precise way but let me hold off on that a little bit okay let me do one other example where we can actually convince ourselves other than just by looking at the graph we can do 9:31 some algebra let's do another example where we convince ourselves that this statement is truly true (truly true?) also true suppose we have some function let's call it G of X which looks like 1 minus x divided by 1 minus the square root of x and i want to know what does G look like 10:02 near x=1 it's not defined at x equals 1 g 1 1-1 / 1-1 which is 0 / 0 and this is a problem g equals want one is not in the domain of this function this function is defined for all positive numbers except well actually all 10:32 non-negative numbers except for x equals 1 so how can we figure this out again we can draw a graph that's one thing we can do and so if i asked my computer to draw the graph it draws me a nice graph of this function when x is 0 we get 1 over 1 which is one and then as x increases the function increases here at x equals 1 we don't have a value and then it 11:03 carries on nicely after that and it looks kind of like a square root function and this value here if we plug in a bunch of numbers let me not you can we plug in a bunch of numbers will see that this gets very close to two it's not too because it's not defined when x is one but it's close to two now I want to claim here that in fact the limit as X goes to one of G of X is in fact too 11:38 and I want to do more than just say it's true saying is true is fine you know what this means now I hold means we know that it gets really close here but let's confirm that it's true so here this to confirm that this is true i'm going to do some algebra with this and check if I can do some algebraic simplification to make this 12:04 look a lot like something that I can do so if i take 1 minus x + 1 minus square root of x so this is G of X it's this and here i'm going to assume X is not one ever forbid so i'm going to assume this X is never one this is my hypothesis here because I want to calculate the limit as X is near one but 12:35 X is never allowed to be one so i can assume X is not one I take this and I'm going to do some algebra and I don't like this square root in the bottom so what I'm going to do to this is I'm going to do a magic trick x 1 minus 1 plus the square root of that over 1 plus the square root of x that's just one back to make it clear it's just I'm going to x 15 x 1 nothing has 13:03 changed but i'm going to write one in a weird way i'm going to write one as 1 plus the square root of x divided by the square of X X is not one well it doesn't matter here and now when i do the algebra out on the top i get 1 plus x times 1 is 1 sorry 1 minus X plus square root of x minus x squared x 13:40 divided by 1 and the middle term drops out because I get a minus square root X + + square root X and they cancel and then square root of x times square root X is X so now i have to do a little more notice that this is 1 minus X and this 14:05 is the square root of x times 1-x plus the square root of x times 1-x and on the bottom I still have 1-x oops I'm walking off the screen. so i have 1 minus X plus square root of 14:32 x times 1 minus x divided by 1 minus X and i can simplify that by factoring 1-x on each of these terms and on the bottom i still have one X so this is that but this is not one so if X is not 1 1 minus x over 1 minus X make sense 15:08 and it is 1 so that means that this is just 1 plus the square root of x so after all of this algebra blah blah what have we done shown that the following: (this is g(x)) if x is not one then g of x is exactly the same as 1 plus the square root of x so important 15:41 to remember as long as X is not one these are the same function as long as X is not one but this one's easy now that means that if we take the limit as X goes to one of G of X that will be exactly the same thing as the limit as X goes to 1 of 1 plus the square root of x 16:04 because they are exactly the same as long as X is not one and when we're taking women we don't care what happens at the value only here this is just too so that means that we've actually proven we've actually made sure that the limit as X goes to one of G of X 2 ok they're not always this hard. I picked a 16:44 hard one on purpose to show you that we can do hard ones.