Limits intro (video) | Limits and continuity | Khan Academy (2024)

Video transcript

In this video, I wantto familiarize you with the idea of a limit, whichis a super important idea. It's really the idea that allof calculus is based upon. But despite beingso super important, it's actually a really, really,really, really, really, really simple idea. So let me draw afunction 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 minus1 over x minus 1. And you might say,hey, Sal look, I have the same thing in thenumerator and denominator. If I have somethingdivided by itself, that would just be equal to 1. Can't I just simplifythis to f of x equals 1? And I would say, well,you're almost true, the difference betweenf of x equals 1 and this thing right over here,is that this thing can never equal-- this thing isundefined when x is equal to 1. Because if you set,let me define it. Let me write it overhere, if you have f of, sorry not f of 0, if youhave f of 1, what happens. In the numerator,we get 1 minus 1, which is, let me just writeit down, in the numerator, you get 0. And in the denominator, youget 1 minus 1, which is also 0. And so anything divided by0, including 0 divided by 0, this is undefined. So you can makethe simplification. You can say that this is you thesame thing as f of x is equal to 1, but you would have to addthe constraint that x cannot be equal to 1. Now this and thisare equivalent, both of these aregoing to be equal to 1 for all other X's otherthan one, but at x equals 1, it becomes undefined. This is undefined andthis one's undefined. So how would Igraph this function. So let me graph it. So that, is my y isequal to f of x axis, y is equal to f of x axis,and then this over here is my x-axis. And then let's say this isthe point x is equal to 1. This over here would bex is equal to negative 1. This is y is equal to 1, rightup there I could do negative 1. but that matter much relative tothis function right over here. And let me graph it. So it's essentially forany x other than 1 f of x is going to be equal to 1. So it's going tobe, look like this. It's going to looklike this, except at 1. At 1 f of x is undefined. So I'm going to put alittle bit of a gap right over here, the circle to signifythat this function is not defined. We don't know what thisfunction equals at 1. We never defined it. This definition of thefunction doesn't tell us what to do with 1. It's literally undefined,literally undefined when x is equal to 1. So this is the functionright over here. And so once again, if someonewere to ask you what is f of 1, you go, and let's say thateven though this was a function definition, you'd go,OK x is equal to 1, oh wait there's a gap inmy function over here. It is undefined. So let me write it again. It's kind of redundant, but I'llrewrite it f of 1 is undefined. But what if I wereto ask you, what is the functionapproaching as x equals 1. And now this is starting totouch on the idea of a limit. So as x gets closerand closer to 1. So as we get closerand closer x is to 1, what is thefunction approaching. Well, this entiretime, the function, what's a gettingcloser and closer to. On the left hand side,no matter how close you get to 1, as longas you're not at 1, you're actually at fof x is equal to 1. Over here from the right handside, you get the same thing. So you could say, andwe'll get more and more familiar with this ideaas we do more examples, that the limit as x andL-I-M, short for limit, as x approaches 1 of f of xis equal to, as we get closer, we can get unbelievably, wecan get infinitely close to 1, as long as we're not at 1. And our function isgoing to be equal to 1, it's getting closer andcloser and closer to 1. It's actually at1 the entire time. So in this case, wecould say the limit as x approaches1 of f of x is 1. So once again, it has very fancynotation, but it's just saying, look what is afunction approaching as x gets closerand closer to 1. Let me do another example wherewe're dealing with a curve, just so that you havethe general idea. So let's say thatI have the function f of x, let me just forthe sake of variety, let me call it g of x. Let's say that we haveg of x is equal to, I could define it this way, wecould define it as x squared, when x does not equal, I don'tknow when x does not equal 2. And let's say that when xequals 2 it is equal to 1. So once again, a kindof an interesting function that, as you'llsee, is not fully continuous, it has a discontinuity. Let me graph it. So this is my yequals f of x axis, this is my x-axisright over here. Let me draw x equals 2, x,let's say this is x equals 1, this is x equals 2, this isnegative 1, this is negative 2. And then let me draw, soeverywhere 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,looks something like this, let me draw a betterversion of the parabola. So it'll looksomething like this. Not the most beautifullydrawn parabola in the history ofdrawing parabolas, but I think it'llgive you the idea. I think you know what aparabola looks like, hopefully. It should be symmetric,let me redraw it because that's kind of ugly. And that's looking better. OK, all right, there you go. All right, now, this would bethe graph of just x squared. But this can't be. It's not x squaredwhen x is equal to 2. So once again, whenx is equal to 2, we should have a little bitof a discontinuity here. So I'll draw a gap right overthere, because when x equals 2 the function is equal to 1. When x is equal to2, so let's say that, and I'm not doing them on thesame scale, but let's say that. So this, on the graph of fof x is equal to x squared, this would be 4, this wouldbe 2, this would be 1, this would be 3. So when x is equal to 2,our function is equal to 1. So this is a bit ofa bizarre function, but we can define it this way. You can define a functionhowever you like to define it. And so notice, it'sjust like the graph of f of x is equal to x squared,except when you get to 2, it has this gap,because you don't use the f of x is equal to xsquared when x is equal to 2. You use f of x--or I should say g of x-- you use gof 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 then at 2, justat 2, just exactly at 2, it drops down to 1. And then it keeps goingalong the function g of x is equal to, or Ishould say, along the function x squared. So my question to you. So there's a coupleof things, if I were to just evaluatethe function g of 2. Well, you'd lookat this definition, OK, when x equals 2, I usethis situation right over here. And it tells me, it'sgoing to be equal to 1. Let me ask a moreinteresting question. Or perhaps a moreinteresting question. What is the limit as xapproaches 2 of g of x. Once again, fancy notation,but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer andcloser to 2, as you get closer and closer, and this isn'ta rigorous definition, we'll do that in future videos. As x gets closer and closer to2, what is g of x approaching? So if you get to 1.9, andthen 1.999, and then 1.999999, and then 1.9999999, whatis g of x approaching. Or if you were to go fromthe positive direction. If you were to say2.1, what's g of 2.1, what's g of 2.01, what's g of2.001, what is that approaching as we get closerand closer to it. And you can see it visuallyjust by drawing the graph. As g gets closerand closer to 2, and if we were tofollow along the graph, we see that weare approaching 4. Even though that's notwhere the function is, the function drops down to 1. The limit of g of x as xapproaches 2 is equal to 4. And you could even do thisnumerically using a calculator, and let me do that, because Ithink that will be interesting. So let me get thecalculator out, let me get my trusty TI-85 out. So here is my calculator,and you could numerically say, OK, what's itgoing to approach as you approach x equals 2. So let's try 1.94,for x is equal to 1.9, you would use this topclause right over here. So you'd have 1.9 squared. And so you get 3.61, well whatif you get even closer to 2, so 1.99, and once again,let me square that. Well now I'm at 3.96. What if I do 1.999,and I square that? I'm going to have 3.996. Notice I'm goingcloser, and closer, and closer to our point. And if I did, if Igot really close, 1.9999999999 squared,what am I going to get to. It's not actuallygoing to be exactly 4, this calculator justrounded things up, but going to get to a numberreally, really, really, really, really, really, really,really, really close to 4. And we can do something fromthe positive direction too. And it actually hasto be the same number when we approach from the belowwhat we're trying to approach, and above what we'retrying to approach. So if we try to 2.1squared, we get 4.4. If we do 2. let me go a coupleof steps ahead, 2.01, so this is muchcloser to 2 now, squared. Now we are gettingmuch closer to 4. So the closer weget to 2, the closer it seems likewe're getting to 4. So once again,that's a numeric way of saying that thelimit, as x approaches 2 from either direction of gof x, even though right at 2, the function is equal to 1,because it's discontinuous. The limit as we'reapproaching 2, we're getting closer, andcloser, and closer to 4.