WEBVTT



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Okay, so today we're talking about limits.

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What is a limit?

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this okay you don't have to write this

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first sentence down actually you don't really you don't need to write this down

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yet so I put it in there my calculator some function you don't know what it is

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secret I put into y1 and here's what I want to know I want to know what is our

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function approaching as X approaches the number 2 okay I don't know what it looks

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like but as as we plug in values closer and closer to X what are these values

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getting close to? What is this number? Okay, and as we approach from the other side,

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what

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is this approaching? Okay, so this is the whole idea of a limit is as x gets closer

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to some number, what is our function value getting closer to? Okay, and this is the

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notation

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we use. LIM stands for limit. And this is x, we have a little arrow always

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approaches

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some number, we put a number here. And sometimes you'll see a positive or a negative

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sign,

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and also so that means, and of this function.

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So we wanna know what x approaches two,

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as x approaches two,

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so what numbers should I plug into my calculator?

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Kind of like we did on the quiz.

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We kept plugging in numbers closer and closer to 4.

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So let's try like 1.9.

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1.99.

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1.999.

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So I'm plugging in closer x values closer and closer to 2, right?

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And what does it look like after x is getting closer and closer and closer to 2?

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7.

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OK.

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So whatever this function is, we think

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this is getting closer and closer to 7,

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even though we do not know what happens at x = 2.

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For the limit, we do not care what happens exactly at x = 2.

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All we care is what's happening as we get closer to it.

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So what we actually just found, where's my cap?

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We actually only approached it from the left side,

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numbers less than two.

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Okay, we also need to approach it from the right side,

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numbers greater than two.

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So I'm gonna plug in, oops.

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Okay, so as we plug the numbers from the right, the numbers are slightly greater

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than two

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and the closer we got to two, what is our function getting closer and closer to

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seven

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as well?

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Okay, limits, we don't care what happens at this number two, we only care what's

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happening

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from the left and from the right.

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Okay, so whatever this function is, and so we use certain notation.

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If we approach from the left, we put a little, it looks like to the power of a

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negative sign.

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Okay, this is me, we're approaching x is approaching 2, but number is less than 2 of

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this function.

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Okay, and the answer was 7.

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If we approach from the right side, it's called the right-hand limit, this was also

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7.

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Okay, so limits, all we care about is what's happening as x gets closer to some

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number,

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both from the right and from the left.

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If these both are the same number, this limit, which is just approaching 2, not from

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the

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left, not from the right, but both sides, is equal to 7 as well.

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So this is called a left-hand limit.

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It's called the right-hand limit.

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limit, okay, and if the limit as x approaches whatever number this is, from the left

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equals

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the limit as x approaches a from the right, then the limit as x approaches a, which

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is

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from the left and right.

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Okay, so if these are both some number let's call it L the limit then this is also

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equal to L

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And we do not care what f(2) is.

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And we find limits we don't care what f of that of this value is. All we care

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about is what's happening from the left and from the right. In fact, what is f of 2

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here? It does not even exist. That is okay—we do not care about f(2) while finding

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this limit.

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Later, we will care because the function value helps tell us whether the function is

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continuous.

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For now, one way to find a limit is to plug in values just to the left and right.

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we're finding limits one way is this way you keep plugging in values if you're

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we're finding limits one way is this way you keep plugging in values if you're

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given the equations just the left of it just the right of it and they keep

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getting closer to some number that is the limit what was this secret function

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I think I know.

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No idea.

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Let's see what it is.

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But what kind of function was this?

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What do you call this kind of function?

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Rational!

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Thank you!

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What is a rational function?

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A rational function is a ratio of two polynomials.

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For example, x squared over e to the x is not a rational function. It is not just a

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ratio of functions or expressions. It must be a ratio of two

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polynomials. Thank you. As long as both are polynomials, it is called a

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rational function. As a review, rational functions have one of two

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things they either have a vertical asymptotes or what's the other option

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Oh, well, where is this?

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Okay, so when values that make the denominator zero will either be vertical

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asymptote or

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a hole in the graph.

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Thank you.

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Okay.

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Does this function have a hole in the graph or a vertical asymptote?

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A hole.

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A hole.

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Okay.

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At two, it doesn't look like this.

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It looks like a hole in the graph.

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How can we tell from here that it has a hole?

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That's a hole in the graph.

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If this wasn't here, it would be a vertical asymptote.

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What would happen, in fact, let me put that in the calculator.

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OK, so now it no longer has a hole.

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There's going to be a vertical asymptote at x equals 2.

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What's going to happen when I plug in numbers close to two now?

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Any guesses?

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Okay, what's happening with the y-value?

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Just keeps getting larger except it's negative, so it's really getting smaller and

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smaller

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smaller and smaller and smaller.

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Okay, in fact, and then what's if I plug in numbers greater than two?

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I like 2.01.

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2.001.

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2.001.

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It just keeps getting larger and larger and larger.

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OK, so if this happens on the calculator,

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if you're given a secret function,

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you're plugging it back.

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Or you could plug in the number.

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It doesn't matter.

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Let's talk about that limit, actually.

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Ch-ch-ch-ch.

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So this is my function.

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Was it x squared plus 3?

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No—minus 3.

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So the function is (x squared minus 3) divided by (x minus 2).

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Okay, and I can either write f of x here or I can write in the whole thing

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Okay, so we plug in values they kept going

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Smaller or smaller so we say

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The left-hand limit is negative infinity.

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Some people say the limit does not exist here, but for this one-sided limit we write

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negative infinity. Now, what is the two-sided limit

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as x approaches 2 of this function. If this does not equal this, it does not exist.

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So today we're only talking about limits based on graphs of a function or based on

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tables

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of values. The last example is table values get plugging in numbers or you're

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given the tables of the X and the Y values. Friday we'll start learning based

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on equation or yeah function equations. Alright first question what is the limit

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or let me write down here.

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The limit as x approaches 2 from the left g of x.

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So as x gets closer and closer to 2 from the left, what is this getting closer and

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closer

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and closer to?

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3.

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so this limit equals 3 the limit as X approaches 2 from the right as X gets

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closer and closer to f of x gets closer and closer to 1. And what is the limit as x

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approaches

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2? It does not exist. We don't have to look at anything except this number does not

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equal

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this number, the left-hand limit does not equal the right-hand limit, the limit as x

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approaches 2 does not exist.

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And what is g of 2?

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Where is the red dot at 2?

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There is none.

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So this does not exist as well.

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So at every x value, we always have a left-hand limit,

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a right-hand limit, the limit,

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and the actual function value.

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And they all might be a number, they all might not exist.

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So let's look at five here.

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The limit as x approaches five from the left.

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As x approaches 5 from the left, these values get closer and closer to 2.

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This is the limit as x approaches 5 from the right.

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as x approaches 5 from the right or g of x approaches 2.

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So what is the limit as x approaches 2?

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5?

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1.

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This?

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2.

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Sorry, 2.

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My bad.

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Five. My bad.

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Since the left-hand limit

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equals the right-hand limit, so the two-sided limit is exactly 2.

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And what is G of five?

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One.

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Okay.

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Is our function discontinuous at 5?

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Yeah, definitely.

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OK. And how can we tell?

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Well, obviously we look at it. But if all I gave you is this information,

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the way we tell if it's continuous is does this equal the limit?

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OK, if the function value equals the limit,

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the function is continuous. If the function value does not equal the limit, it is

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discontinuous.

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And then our previous example

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In our previous example, the function value did not exist, so it was definitely

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discontinuous.

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But this doesn't equal well this doesn't exist either, so

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But in fact that's going to be the definition of

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A function is continuous at a point when the limit equals the function value.

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We'll define that later.

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Okay, let's do another one.

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What is the limit as x approaches 0 from the left?

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So as x approaches zero, f of x gets closer and closer to, let's call it 2.5.

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The limit as x approaches zero from the right, as x approaches zero from the right,

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f of

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x gets closer and closer to 2.5.

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What is the limit as x approaches zero?

00:18:54.340 --> 00:19:00.740
2.5. 2.5.

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And what is g of 0? 2.5.

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2.5. Is it continuous at 0?

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Yeah. Okay.

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And in fact, if we know the function is continuous,

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Let's say a 2 and 5 are its only discontinuities.

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Or let me ask you this.

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the limit as x approaches zero,

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x to the fifth minus seven x plus four.

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Is this a continuous function?

00:19:59.380 --> 00:20:00.980
What do we call this type of function?

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Polynomial.

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A polynomial.

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Okay.

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All polynomials are continuous over all real numbers.

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Okay.

00:20:12.580 --> 00:20:19.375
Since we know it's continuous over all real numbers, we know this limit exactly

00:20:19.375 --> 00:20:20.125
equals

00:20:19.860 --> 00:20:21.640
whatever f of zero is.

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Let's call this f.

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And what is f of zero?

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It will be four.

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Okay.

00:20:29.940 --> 00:20:34.340
So if a function is continuous at a value—or over all values—

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the limit as x approaches that value equals the function value.

00:20:40.860 --> 00:20:42.220
okay, so

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Like at four here

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It's definitely continuous there so the limit as X approaches four from the left

00:20:59.980 --> 00:21:04.040
is going to equal the limit as x approaches 4 from the right.

00:21:07.320 --> 00:21:10.800
It's going to equal the limit as x approaches 4.

00:21:14.500 --> 00:21:17.180
It's going to equal g of 4, whatever that number is.

00:21:18.460 --> 00:21:21.260
Let's call it 1.8. I'm just making up a number.

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If the function is continuous, the left-hand limit equals the right-hand limit,

00:21:26.360 --> 00:21:28.060
the limit equals the function value.

00:21:31.220 --> 00:21:34.740
So every other number here, besides two and five,

00:21:34.920 --> 00:21:36.760
the limit from the left will equal the limit from the right.

00:21:37.180 --> 00:21:40.100
The limit will equal the function value.

00:21:41.220 --> 00:21:42.120
Make sense so far?

00:21:42.820 --> 00:21:44.000
Okay, if you're getting a graph, it's real easy.

00:21:44.040 --> 00:21:48.020
Okay, in fact, actually let's do this example.

00:21:51.920 --> 00:21:54.960
What is the limit as x approaches, let's go to two.

00:21:55.280 --> 00:21:56.180
Oh no, let's go one.

00:21:56.060 --> 00:22:02.420
The limit as x approaches 1 from the left is 1.

00:22:03.280 --> 00:22:04.640
It keeps getting closer and closer to 1.

00:22:05.280 --> 00:22:08.280
As x approaches 1 from the right, 1.

00:22:09.240 --> 00:22:11.180
What is the limit as x approaches 1?

00:22:12.200 --> 00:22:13.100
1.

00:22:13.280 --> 00:22:14.700
What is f of 1?

00:22:15.400 --> 00:22:16.300
1.

00:22:16.560 --> 00:22:17.460
No.

00:22:18.460 --> 00:22:19.360
Ok.

00:22:19.900 --> 00:22:22.100
We have a hole in the graph right here.

00:22:23.680 --> 00:22:29.233
You can never tell on Desmos or the graphing calculator there's a hole so be real

00:22:29.233 --> 00:22:29.983
careful

00:22:30.720 --> 00:22:35.220
Okay, because it's so small no matter how far I zoom in here

00:22:36.220 --> 00:22:37.520
Where is it?

00:22:38.080 --> 00:22:40.100
You will never see the hole

00:22:41.240 --> 00:22:43.620
Okay, so be real careful on

00:22:44.720 --> 00:22:46.340
graphing calculators on Desmos

00:22:46.900 --> 00:22:48.900
You will never see holes in the graph

00:22:50.200 --> 00:22:52.900
Okay, that's why when we draw them

00:22:53.960 --> 00:23:02.500
Like here we make the whole giant so, you know, it's a whole that makes sense. Okay

00:23:03.440 --> 00:23:05.280
You don't have to draw this

00:23:06.200 --> 00:23:07.340
We'll just do it real quick.

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What is the limit as this is 0, 0, right?

00:23:20.900 --> 00:23:25.120
As x approaches 0 from the left, let's call this f.

00:23:28.580 --> 00:23:31.960
As x approaches 0 from the left, f of x approaches?

00:23:33.320 --> 00:23:34.220
Maybe 1.

00:23:33.800 --> 00:23:39.360
What is the limit as x approaches 0 from the right?

00:23:42.240 --> 00:23:45.920
As x approaches 0, this approaches negative 2.

00:23:47.840 --> 00:23:49.660
So both of these limits exist.

00:23:50.360 --> 00:23:51.820
Anytime you get a number it exists.

00:23:53.380 --> 00:23:57.340
And what is the limit as x approaches 0?

00:23:59.020 --> 00:24:02.040
It does not exist.

00:24:02.980 --> 00:24:03.880
Why?

00:24:03.260 --> 00:24:09.489
because the left-hand limit does not equal the right-hand limit. Okay and what is F

00:24:09.489 --> 00:24:10.320
of 0?

00:24:12.760 --> 00:24:21.400
negative 1. Okay is it discontinuous at X equals 0? Definitely, that's some sort of

00:24:21.400 --> 00:24:28.436
jump discontinuity we call it. We'll talk later, we actually say it's continuous

00:24:28.436 --> 00:24:30.060
from the left

00:24:30.060 --> 00:24:34.700
because the left-hand limit equals the function value.

00:24:35.520 --> 00:24:41.587
It's discontinuous from the right because the limit from the right does not equal

00:24:41.587 --> 00:24:42.337
the

00:24:42.020 --> 00:24:42.920
function value.

00:24:43.140 --> 00:24:45.920
But overall it's definitely discontinuous at zero.

00:24:46.080 --> 00:24:51.400
What is the limit as x approaches 2 from the left?

00:24:56.160 --> 00:24:57.060
2.

00:24:58.440 --> 00:25:01.620
The limit as x approaches 2 from the right?

00:25:03.420 --> 00:25:04.320
0.

00:25:05.260 --> 00:25:15.660
The two-sided limit as x approaches 2 does not exist because the left-hand

00:25:15.660 --> 00:25:24.020
limit does not equal the right-hand limit. And f(2) = 1.

00:25:25.980 --> 00:25:32.760
It's definitely discontinuous at 2. Is it continuous on the left? No because the

00:25:32.760 --> 00:25:35.800
left-hand limit does not equal the function value.

00:25:36.080 --> 00:25:37.080
Is it continuous from the right?

00:25:37.660 --> 00:25:40.460
No, because the right-hand limit,

00:25:40.620 --> 00:25:42.820
zero does not equal to the function value.

00:25:43.900 --> 00:25:44.800
Okay.

00:25:44.280 --> 00:25:46.600
Unlike at x = 0, it is not continuous from the left.

00:26:00.580 --> 00:26:05.760
What is the limit as x approaches 4 from the left?

00:26:10.880 --> 00:26:11.960
This gets closer to 3.

00:26:13.180 --> 00:26:16.080
3, limit as x approaches 4 from the right.

00:26:17.340 --> 00:26:18.540
3, 3.

00:26:19.640 --> 00:26:21.680
What is the limit as x approaches 4?

00:26:22.300 --> 00:26:23.200
3.

00:26:24.840 --> 00:26:26.640
And what is f of 4?

00:26:26.960 --> 00:26:27.860
3.

00:26:27.900 --> 00:26:28.800
3.

00:26:29.300 --> 00:26:30.200
Okay.

00:26:31.080 --> 00:26:33.240
Is the function continuous at 4?

00:26:33.720 --> 00:26:34.620
Yes.

00:26:34.400 --> 00:26:35.300
Yes.

00:26:34.700 --> 00:26:41.420
If we weren't given a graph, if this limit equals the function value, it's

00:26:41.420 --> 00:26:42.170
continuous.

00:26:43.640 --> 00:26:45.240
Okay, and we're going to define later.

00:26:45.340 --> 00:26:49.520
So that means it's continuous at a point, but later it's continuous on an interval.

00:26:51.400 --> 00:26:53.540
Okay, I'm hoping you're getting the point.

00:26:54.000 --> 00:26:56.900
Let's do, I think I got one more for you.

00:26:57.160 --> 00:26:59.080
Anybody know what function this is?

00:27:00.760 --> 00:27:01.660
This side.

00:27:01.740 --> 00:27:04.140
This is tangent.

00:27:09.280 --> 00:27:14.780
What is the limit as x approaches pi from the left?

00:27:17.920 --> 00:27:18.820
Zero.

00:27:19.960 --> 00:27:22.680
What is the limit as x approaches pi from the right?

00:27:23.320 --> 00:27:24.220
Zero.

00:27:26.360 --> 00:27:28.860
What is the limit as x approaches pi?

00:27:29.720 --> 00:27:30.620
Zero.

00:27:30.320 --> 00:27:31.220
Zero.

00:27:31.160 --> 00:27:37.380
As x approaches pi over 2 from the left, the tangent values grow larger and larger.

00:27:37.380 --> 00:27:38.280
and larger.

00:27:39.120 --> 00:27:41.040
So we put positive infinity.

00:27:48.300 --> 00:27:52.720
Now consider the limit as x approaches pi over 2 from the right.

00:27:57.060 --> 00:28:02.300
It approaches negative infinity; from the right, the graph keeps going down

00:28:02.300 --> 00:28:13.280
forever. The two-sided limit as x approaches pi over 2 does not exist

00:28:13.280 --> 00:28:19.260
because the one-sided limits are not equal. If both approached positive infinity, we

00:28:19.260 --> 00:28:20.180
would write

00:28:20.180 --> 00:28:21.960
positive in here, they both go down.

00:28:22.780 --> 00:28:23.880
But they go in different direction,

00:28:24.220 --> 00:28:25.120
definitely doesn't exist.

00:28:32.560 --> 00:28:34.800
So is it continuous at pi over two?

00:28:37.980 --> 00:28:41.560
Definitely not, in fact, what is tan of pi over two?

00:28:44.960 --> 00:28:45.960
Does not exist.

00:28:47.940 --> 00:28:48.840
Okay.

00:28:50.180 --> 00:28:53.220
What do we call this type of discontinuity?

00:28:56.440 --> 00:28:59.540
We just call it infinite discontinuity.

00:29:01.100 --> 00:29:05.240
A hole in the graph we call a removable discontinuity.

00:29:07.160 --> 00:29:09.200
If it jumps, it's called a jump discontinuity.

00:29:09.260 --> 00:29:10.180
OK.

00:29:11.360 --> 00:29:14.940
This is a function of sine of pi over x.

00:29:21.360 --> 00:29:22.260
OK.

00:29:21.600 --> 00:29:24.500
It's continuous everywhere except to what?

00:29:26.080 --> 00:29:31.480
At x = 0 the function is undefined, but for every other x we can evaluate sine of

00:29:31.480 --> 00:29:32.230
that value.

00:29:33.500 --> 00:29:36.140
The interesting part the limit has

00:29:37.620 --> 00:29:39.740
X approaches zero

00:29:44.260 --> 00:29:48.180
Okay to get closer to closer to zero what's happening

00:29:51.920 --> 00:29:54.260
So let's look at this as x goes to zero,

00:29:55.320 --> 00:29:57.140
what's happening to this?

00:29:59.520 --> 00:30:02.000
What's pi over a very, very small number?

00:30:04.520 --> 00:30:09.300
Like pi divided by 0.00001 is a very big number.

00:30:10.680 --> 00:30:13.780
The closer x gets to zero, the larger pi divided by x becomes in magnitude.

00:30:15.680 --> 00:30:18.100
And what is sine of a very, very big number?

00:30:21.120 --> 00:30:23.040
It just keeps doing this forever.

00:30:23.240 --> 00:30:25.560
It keeps oscillating.

00:30:26.400 --> 00:30:31.360
As x approaches zero, the inside approaches infinity in magnitude, and the

00:30:31.360 --> 00:30:33.180
inside just keeps oscillating back and forth.

00:30:34.000 --> 00:30:34.980
You get this really cool path.

00:30:35.440 --> 00:30:41.760
So this, both from the left and from the right, don't approach a single number.

00:30:42.600 --> 00:30:46.320
It never exceeds 1 or goes below negative 1, but

00:30:46.320 --> 00:30:49.020
It does not approach a number, so this does not exist.

00:30:51.020 --> 00:30:53.540
It's called an oscillating discontinuity.

00:30:54.480 --> 00:30:57.180
It's actually really cool if you look at the on Desmos.

00:31:20.720 --> 00:31:24.500
The more I zoom in, the more there are.

00:31:27.520 --> 00:31:29.380
It completely fills up the whole screen.

00:31:30.540 --> 00:31:33.780
In reality, there is a tiny gap between every one of those lines.

00:31:35.080 --> 00:31:38.080
But the closer you get, the faster it goes back and forth, back and forth.

00:31:39.280 --> 00:31:40.180
Super cool.

00:31:42.320 --> 00:31:45.200
Okay, very weird as well.

00:31:46.180 --> 00:31:50.380
But there we go.

00:31:50.780 --> 00:31:53.500
Okay I think we're going to stop here.

00:31:57.620 --> 00:32:02.709
Again all the limits, the idea of left-hand limit, right-hand limit, the limits,

00:32:02.709 --> 00:32:03.459
it's

00:32:03.210 --> 00:32:08.390
only based on tables of values and graphs and Friday we'll talk about when

00:32:08.390 --> 00:32:10.430
we are given the actual equation.

00:32:11.250 --> 00:32:15.690
We're just going to, I put a new Delta Math assignment up, so just do Delta Math.

00:32:16.930 --> 00:32:17.830
So you can get going on that.