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# Limits intro

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.A (LO)
,
LIM‑1.A.1 (EK)
,
LIM‑1.B (LO)
,
LIM‑1.B.1 (EK)

## Video transcript

in this video I want to familiarize you with the idea of a limit which is a super important idea it's really the idea that all of calculus is based upon but despite being so super important it's actually a really really really really simple idea so let me draw a function here actually let me define a function here a kind of a simple function so let's define f of X let's say that f of X is going to be X minus 1 over X minus 1 and you might say hey Sal look I have the same thing in the numerator in the denominator if I have something divided by itself that would just be equal to 1 can't I just simplify this to f of X equals 1 and I would say well you're almost true the difference between f of X equals 1 and this thing right over here is that this thing can never eat this thing is undefined when X is equal to 1 because if you set let me define it right let me write it over here if you have F of sorry not F of 0 if you have F of 1 what happens in the numerator you get 1 minus 1 which is let me just write it down in the numerator you get 0 and then the denominator you get 1 minus 1 which is also 0 and so anything divided by 0 we're including 0 divided by 0 this is undefined this is undefined so you can make the simplification you can say that this is you can say that this is the same thing as f of X is equal to 1 but you would have to add the constraint that X cannot X cannot be equal to 1 now this and this are equivalent both of these are going to be equal to 1 for all other X's other than 1 but at x equals 1 it becomes undefined this is undefined and this one's undefined so how would i graph this function so let me graph it so that is my y is equal to f of x axis y is equal to f of x axis and then this over here is my x axis x axis and then let's say that this is the point X is equal to one this over here would be X is equal to X is equal to negative one this is y is equal to one right up there I could do negative one with that matter much relative to this function right over here and let me graph it so it's essentially for any X's for any X other than one f of X is going to be equal to one so it's going to be look like this it's going to look like this except at one at one f of X is undefined so I'm going to put a little bit of a gap right over here the circle to signify that this function is not defined we don't know what this function equals at one we never defined it this definition of the function doesn't tell us what to do with one it's literally undefined literally undefined when x is equal to one so this is this is the function right over here and so once again if someone were to ask you what is f of one you'd go see it and let's say that even if this was a function definition you would go KX is equal to 108 there's a gap in my function over here it is undefined so let me write it again I'm that's why it's kind of redundant but I'll rewrite it F of one is undefined but whatever what if I were to ask you what is the function approaching as x equals one and now this is starting to touch on the idea of a limit so as X gets closer and closer to one so as we get closer and closer X's to one as we get closer and closer X's to one what is the function approaching well this entire time the function what's it getting closer and closer to on the left-hand side no matter how close you get to one as long as you're not at one you're actually at f of X is equal to one over here from the right hand side you get the same thing so you could say and you'll will get more and more familiar with this idea as we do more examples that the limit as X and L I am short for limit as X approaches 1 as X approaches 1 of f of X of f of X is equal to as we get closer we can get unbelievably we can get infinitely close to one as long as we're not at one and our function is going to be equal to one it's getting closer and closer and closer to one it's actually at one the entire time so in this say k case we could say the limit as X approaches 1 of f of X is 1 so once again has very fancy notation we're just saying look what is the function approaching as X gets closer and closer to 1 let me do another example where we're dealing with a curve just so that you have the general idea so let's say let's say that I have the function f of X let me just just for the sake of variety let me call it G of X let's say that we have G of X is equal to I could define it this way we could define it as x squared when when X does not equal I don't know when X does not equal 2 and let's say that when x equals 2 when x equals 2 it is equal to 1 so once again it kind of an interesting function that as you'll see is is not fully continuous it has a discontinuity so let me graph it so this is my y equals f of X axis this is my x axis right over here let me draw x equals 2x let's say this is x equals 1 this is x equals 2 this is negative 1 this is negative 2 and then let me draw so everywhere everywhere except x equals 2 it's equal to x squared so let me draw it like this so it's going to be a parabola look something like this it's going to look something let me draw a better version of the parabola so it'll look something like this it looks something like this not the most beautifully drawn parabola in the history of drawing parabolas but I think it'll give give you the idea I think you know the parabola looks like hopefully it's a would other way it should be symmetric it let me draw a redraw it because that's kind of ugly okay me and that's looking better okay all right there you go all right now this is this would be the graph of just x squared but this can't be it's not x squared when X is equal to two so once again once again when X is equal to two we should have a little bit of a discontinuity here so I'll draw a gap right over there because when x equals two the function is equal to 1 when X is equal to 2 so let's say that and I'm not doing them on the same scale but let's say that so this on X on the graph of f of X is equal to x squared this would be 4 this would be 2 this would be 1 this would be 3 so when X is equal to 2 our function is equal to 1 so when X is equal to 2 our function is equal to is equal to 1 so there's a bit of a bizarre function but we can define it this way you can define a function however you like to define it and so notice it's just like the graph of x of x of f of X is equal to x squared except when you get to 2 it has this gap because that's you don't use the f of X is equal to x squared when X is when X is equal to 2 you use f of X or I should say G of X you use G of X is equal to 1 have I been saying f of X I apologize for that you use G of X is equal to 1 so then the then at 2 we just set to just exactly at 2 it drops down to 1 and then it keeps going along the function G of X is equal to or I should say along the function x squared so my question to you so there's a couple of things if I were to just evaluate the function G of 2 while you look at this definition okay when x equals 2 I use this situation right over here and it tells me it's going to be equal to 1 let me ask a more interesting question or perhaps a more interesting question what is the limit as X approaches 2 of G of X of G of X once again fancy notation but it's asking something pretty pretty pretty simple it's saying as X gets closer and closer to 2 as you get closer and closer and this isn't a rigorous definition we'll do that in future videos as X gets closer and closer to 2 what is G of X approaching so if you get to one point nine and then one point nine nine nine and then one point nine nine nine nine nine nine and then one point nine nine nine nine nine nine nine what is G of X approaching and or if you were to go from the from the positive direction if you were to say two point one what's G of two point one what's G of two point oh one what's G of two point zero zero one what is that approaching as we get closer and closer to it and you can see it visually just by drawing the graph as G gets closer and closer to two and if we were to follow it along the graph we see that we are approaching four even though that's not where the function is the function drops down to one the limit of G of X as X approaches two is equal is equal to four and you can even do this numerically using a calculator and let me let me do that because I think that will be interesting so let me get a calculator out let me get my trusty my trusty ti-85 out let me so here is my calculator and you could numerically say okay what's it going to approach as you approach x equals two so let's try 1.9 4 for X is equal to one point nine you would use this top Clause right over here so you'd have one point nine squared and so you'd get three point six one well what if you get even closer to two so one point nine nine and once again let me square that well now I'm at three point nine six but what if I do one point nine nine nine nine nine nine and I square that and I square that I'm going to get three point nine nine six notice I'm going in closer and closer and closer to our point and if I did if I got really close one point nine nine nine nine nine nine nine nine nine nine squared what am I going to get to it I'm it's not actually going to be exactly for this calculator just rounded things up it's going to get to a number really really really really really really really close to four and we can do something from the positive direction too and it actually has to be the same number when we approach from the from the below what we're trying to approach and above what we're trying to approach so if we try to point one squared we get 4.4 if we do to point let me go a couple of steps ahead to point oh wow so this is much closer to 2 now squared now we're getting much closer to 4 so the closer we get to 2 the closer it seems like we're getting to 4 so once again that's a numeric way of seeing that the limit as X approaches 2 from either direction of G of X even though right at 2 the function is equal to 1 because it's discontinuous the limit as we're approaching 2 we're getting closer and closer and closer to 4