| Start | Limits last time so we'll talk about limits some more.
Until we've had enough limits for one lifetime. Anybody have any questions from last time? No? Many of you are missing. You have your MyMathLab assignment was due I think now. Or an hour ago? You have a new one that will start this week that's due next week. It has 8 problems or 9 problems. It shouldn't be very hard. But it'll go active at about noon I think. |
| 0:31 | There's basically going to be one every week. And I'm going to try to keep them to around 10 to 12 problems.
So I don't overburden you. There you go. So we're doing limits so remember the limit is a key concept of calculus. Now this is sort of an applied calculus course for people who aren't really that excited about the whole thing but some of you are going to be more quantitative than others but the concept of limits is crucial for understanding derivatives. |
| 1:02 | So what happened was people had this issue with limits.
Beforehand they said you can't actually get to a number. And what the mathematicians did is they figured out you can act as if you are at the number if you just treat it as a limit. In other words when you look at say the limit when x approaches 5 or a number but you're infinitesimally close to 5. You're so close to 5 that you can pretend it's at 5 and see what happens in the problem. |
| 1:31 | And I think by about Wednesday we'll start to work with that.
So last time we were doing limits with numbers so just to review suppose you had a function and it's the following. |
| 2:07 | This is some invented function. We call it a piecewise function, it's broken into pieces.
And it says when x<1 you use this function, when x=1 you get 0 and when x>1 you use this function. I can't imagine a scenario where this particular piecewise function would show up in life |
| 2:30 | but you see piecewise functions all the time. You know if something like your first 500 text messages are $10 and then they're 2 cents a message after that.
So you really have 2 different functions. You have the before 500 text messages and you have after 500 text messages. So many things in life are 2 versions you know $8 for the first hour of parking, 50 cents for every hour after that, etc. So when we want to represent them mathematically the function has a break at some point. it may not actually break the curve |
| 3:03 | But it has a break. If you were graphing this the question you would ask yourself is what happens at 1?
Well when you're <1 you look like the parabola x^2+5 And you're going to get to about there at (1,6) and you're not equal to 1. When you're actually at 1, x=0. |
| 3:32 | So this is an example of a function that is not continuous.
Which I don't think we do much on continuity but continuous means you don't leave the chalkboard when you're drawing it. There's technical definitions but in other words when you leave the first function you go directly to the second function. So here you don't. Now when x>1 you get the equation 2x^2 + 4 Which would not really be a continuation of this parabola it would be a steeper second half. It doesn't really show. |
| 4:03 | But this first parabola would probably look more like that.
But at 1 you're back to 6. So your only trouble spot is right here at 1. So you say to yourself well if I'm not actually at 1 then I'm just a little bit less than that. So if I approach 1 from the left side just less than 1 I'm going to get 6. |
| 4:32 | And when I approach x from the positive side just greater than 1 again I'm going to get 6.
And that's from plugging in 1. And you can take your calculator and you can plug in 0.9999999 put in a bunch of 9's And you'll get just less than 6 you'll get 5. a bunch of 9's. If you put in just bigger than 1 like 1.0000001 again you'll get just off of 6. In fact if you go far enough the calculator will say 6 because it's only good to about 10 decimal places. |
| 5:06 | But you're not actually. But f(1) = 0.
f(1) does not equal 6. That's why you get that hole there. But if we ask for the limit as x approaches 1 with no minus or plus we would also get 6. And that's because this limit = this limit. |
| 5:31 | So as long as they're equal, as long as the lefthand limit equals the righthand limit then the limit of the number itself exists.
So remember if we had another one like we'll do in a second, if they don't match then the limit does not exist. So if you had f(x) |
| 6:04 | Okay so just change that by one number.
So now you ask yourself what happens when x<1? When x<1 you're gonna get the curve x^2 + 5. And you're going to stop at 6. When x=1 you get 0. |
| 6:32 | When x>1 you get 2x^2 + 3 so now you have this.
So it's doing what's called jumping. It jumped this time. It's slightly different than a hole. Which is called removable discontinuity. Don't worry about the terms. So again so as x approaches 1 form the minus side |
| 7:07 | Is 6 but the limit as x approaches- I'll bring this back down in a second.
The one from the other side is going to be 5. That's because when you plug in 1 into this equation you get 1+5=6 When you plug in 1 into this equation you get 2+3=5 Since they don't match you say the limit when x -> 1, we haven't specified left or right hand, then you say the limit does not exist or as we say DNE. |
| 7:50 | Even though f(1) exists. So you don't want to confuse the limit with the actual value.
|
| 8:01 | And then in calculus we pretend that it is the actual value so we kind of have both brains. Sometimes we say that it's the number, sometimes we say that it's not the number.
It takes a little mathematical work to get those two concepts to match each other. Or work with each other. Alright so now let's make this more annoying because that's what we do in the math world. |
| 8:40 | Alright so now let's make this more fun.
So say we have f(x) = 5/x So 5/x is not a very interesting function right? |
| 9:01 | You plug in a number you get a fraction. There's only 2 interesting things that go on here.
One is what happens at zero. And there's what happens at infinity. When we say infinity with a limit we mean as large as we want x to get. A million, a billion, a trillion, infinity. Okay? There's numbers between a trillion and infinity. In fact there's an infinite number of numbers between a trillion and infinity. But what is happening to this fraction and we make the denominator bigger and bigger? |
| 9:39 | So what is happening as x approaches ∞ ?
We have 5 on top and the denominator is a million. 5/1,000,000 is a very small number. 5/1,000,000,000 is a really small number. 5/1,000,000,000,000 and you keep going so eventually you just get 0. So we can say at ∞ but we can't actually get to ∞ so we can't actually get to 0. In face when you graph this |
| 10:07 | you have a horizontal asymptote which symbolizes that you're getting closer and closer to 0 but you're not actually getting there.
Which you get as we say arbitrarily close to 0. You get as close to 0 as you want. If somebody says can you get to 10 decimal places of 0? You say sure. Just give me a value of x and we'll get you to 10 decimal places. |
| 10:30 | How about 100 decimal places? Sure I can, as many decimal places as you want.
So that's what we mean by arbitrarily close to 0. What about -∞? Remember what -∞ is it's negative one billion, negative a gazillion. Okay? Negative in the other direction. Well you sort of get -0 which doesn't mean anything so you get 0 again. What's because the function is doing this on the other side. |
| 11:03 | So again at 0.
So one of the very important limits to sort of stick in your brain. Is that one. Or to make it more general. |
| 11:48 | If you have a constant on top and you have x on the bottom whether you're going to positive infinity or negative infinity your limits going to be 0.
Just think about that for a second say well sure. |
| 12:01 | I plug numbers in the denominator gets bigger and bigger and we have a fraction the numerator is a number and the denominator is a really big number.
That's basically 0. Say we have a pizza and I say you can have one trillionth of the pizza. You're not getting much of that pizza. Oh I just won the lottery I won $100 million I'll give you one cent. That's not a very big fraction right? So you might as well say well I'll just take zero cents then. And we're not friends anymore right? And by the way if I won $100 million we're not friends. |
| 12:30 | Just to make that clear okay?
You can come visit me on my tropical island. Anyway I'm not going to win it that fast you have to buy a ticket first. The odds are basically 0. It's fun to buy them but you don't really think you're going to win. In fact x can be any power greater than or equal to 1. |
| 13:00 | And that'd still be around 0.
Even if it's less than ??. As long as it's positive. Even if it was a square root. Eventually you'd get there. The square root of a trillion is a million. It's still a big number. So if you have 5/x, 5/x^2, 5/x^3, 5/x^4, 5/x^1/1000, you're going to get to 0 eventually. |
| 13:40 | Okay? So big limit to get the hang of.
Now suppose instead I ask you what happens when you go to 0. What if I'm doing the same equation. Well now it gets a little more interesting. So think about a fraction where you have a very small number in the bottom. So say you have 5/0.000001 |
| 14:09 | So remember what you do when you have a number divided by a fraction. You flip and multiply.
So this is 5 * 1,000,000 which is 5,000,000 which is a lot. So when the person who wins the lottery says I'll just take 1/0.000001 of whatever you want. |
| 14:30 | If they're dumb they'll say sure good deal.
Get them to sign something for that right? So when you divide by a million, it's one millionth, it's multiplying by one million. If this was a billionth it's multiplying by a billion. So it's going to get very very large. Infinitely large. So as x approaches 0 from the plus side you're going to get positive infinity. So that's the second one for our little note board. |
| 15:13 | Again as long as n>0.
So we have a fraction a rational expression. Where the top is just a constant. We'll deal in a little while with when it's an equation as well. |
| 15:32 | But if it's a constant and the bottom is x to a power as you get closer and closer to 0 look at what happens to the graph. It goes up.
What about if that's a negative number? If this was -1,000,000 or -1,000,000,000 that would be negative. So if we did the limit of as x approaches 0 from the minus side of f(x) we would get -1,000,000. So eventually we're going to get to -∞. |
| 16:06 | So that's why you need these sort of left and right hand limits.
Let me know if I'm going to fast. Well when you have 0 right? x^0 is equal to 1 so you just get the constant. |
| 16:36 | Good question though I was waiting for somebody to ask that.
And if it's a negative power then x moves up into the numerator and totally changes it. And if you look at this graph, as x approaches -∞ or as x approaches 0 from the negative side, this way, you go down to -∞. |
| 17:00 | So those two don't match so the limit as x goes to 0 no sign literally does not exist.
So far so good? Alright so we'll store those. |
| 17:46 | So if we had
something like that.
|
| 18:20 | So this is basically what we just wrote. constant on top, x^4 on bottom, x gets bigger and bigger, this becomes 0.
|
| 18:33 | Now I made this one slightly tricky.
Not too tricky though. So if I do the limit as x approaches -∞ of 3/x^4 is also going to be 0 because 0 doesn't really have a sign to it. Positive 0, negative 0, this sort of means something if you put a -0 that would really just be a teeny bit less than 0. |
| 19:01 | But it's still 0.
If I owe you a billionth of a cent you're getting 0. If you owe me a billionth of a cent I'm getting 0. But there is a side to it. But it's not important. However we can mess with that a little bit. |
| 19:33 | So if we did the limit as x approaches 0+ of 3/x^4 that's going to be positive infinity. That's our rule.
And if we do the limit of x approaches 0- 3/x^4 well it turns out that's also positive infinity |
| 20:02 | because x^4 is always positive. This graph looks like this.
This is also positive infinity which means we have to change our rule a little bit. So therefore it doesn't matter. Those two match so don't confuse the limit is infinity with the limit does not exist. So now we have to go back and we have to change this a little bit. |
| 20:35 | And by the way if it has a -3 on top it would get interesting.
We have another assumption here. k is ?? Because if k is 0 it's meaningless right? And this is true if n is a negative number. Then the limit when x approaches 0- of k/x^n is positive infinity when n>0. |
| 21:07 | That's the fun thing about mathematicians you can come up with something then immediately try to find an exception.
Something wrong with your rule. They're very annoying. They're like your little brother whose always trying to find what you're doing wrong. However one way to do it is you start doing test cases and you say wait I found one where it's always positive ∞. So there's two possibilities really with the k/x^n. |
| 21:30 | You get that first graph where it's gonna go down on one side and up on the other side.
Or the second graph where they're both going to go up. Now you can make them both go down, that would just be making k negative. But that's not important. Just these two types are important. How are we going so far? Loving this? Love is a strong word. |
| 22:08 | Play with this a little more.
Alright so what's that limit? |
| 23:13 | Well it's pretty much the same as before right?
The fact that we're subtracting 1 doesn't really have any effect on that. So that'd really still be 0. Because if we have a trillion minus 1 that's still essentially a trillion. Right? Infinity minus 1 is still infinity so that's going to be 0. |
| 23:32 | And if we do -∞ it's going to be 0 alright.
What happens when we go to 0? What do we get? Anyone want to take a shot? |
| 24:00 | Right. We don't get ∞ because it's not interesting at 0, this is just -8.
You plug in 0 you get -1 on bottom you get 8 on top. So the interesting point on this one is at x=1. So now the limit as x approaches 1 from the positive side of 8/x-1 is ∞. Because this graph looks like the other one except we shift at that vertical asymptote. |
| 24:35 | At x=1.
So you have to be careful of what's in the denominator. The basic concept hasn't changed. It's that now the number, instead of approaching 0 it's approaching 1. And if we approached 1 from the other side that would be -∞. |
| 25:04 | And in fact here it doesn't really matter if we approach it from the plus or minus side we're still going to get -8
whether that's a plus or a minus.
It's still going to come out -8 because this is going to be 0-ish, 0, so you have 8/-1 whether that's positive or negative. So you have to be careful. Sometimes you get this instinct and you just say to yourself well I see something on top something on the bottom I'm going to 0 then that makes it ∞. |
| 25:37 | Okay you have to make sure that it's where the positive is 0 that matters.
|
| 26:23 | There's not much more to do with that hm we're getting a little ahead. Well we'll practice something else.
|
| 26:44 | You could have something like that.
|
| 27:00 | ??
Maybe it's safer if I write n>1. Why did I write n<2? I apologize this should say n is odd. |
| 27:30 | And n is even. That'll be fun 5 minutes from now when people are watching this on video.
Maybe they won't be watching this video. I apologize it should say odd and even I don't know why I wrote negative and positive. Okay so now let's look at when we have the square root. So if you plug in just bigger than 5 you're getting √0. What's the √0? Good guess. So you get positive ∞. How about when I approach 5 from the minus side? |
| 28:12 | So now you wanna say -∞ right? But be careful. What's the problem?
You can't take the square root of a negative number or with the reals you can't take the square root of a negative number. So you can't get to negative this only exists for x>5. |
| 28:32 | It doesn't exist when x=5. Remember we did that with domain?
Because you plug in 5 you get 0 in the denominator and you can't have less than 5. So this doesn't even mean anything. This equation doesn't have a left side. It only has a right side. |
| 29:00 | But you can do this with positive ∞.
You plug in an infinitely large number you get √∞. √∞ = ∞. So that's going to approach 0. So far so good? |
| 29:36 | If however I gave you
9 over the cube root of x-5
If I plug in a number just a little bit bigger than 5 I'm going to get positive ∞ again.
|
| 30:02 | Because the cube root of 0 is also 0.
But if I did the cube root of -5 well you can take the cube root of negative numbers. |
| 30:30 | Now you'll get -∞.
So far so good? There's lot of these things so you feel like there's lots of exceptions. But not really. It's just always tricky when you want to write rules. Because you have to think of all the different cases. Or you have to restrict it. So we're going to restrict this to just when n>1. And by the way if this was -9 then this would be -∞ and if this was -9 this would be +∞ because you're just changing the sign of everything. |
| 31:04 | So far so good?
|
| 31:35 | We could do this. Nah let's just wait.
Alright it's Friday. That's enough for one day. I was going to do 5 more minutes but I got kind of ahead of myself so see you on Monday. Thanks for coming in folks. |