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Okay, today we're talking about...

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What are we talking about?

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We're going to have a quiz on Wednesday, not today.

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I want you to do this last thing.

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I want to put a limit as infinity.

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That means either the limit as x approaches infinity or some function, the limit as

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x

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approaches negative infinity.

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Some other way.

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This is the exact same thing we did in math 2.3,

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This is the same idea you studied as the end behavior of a function.

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It's the exact same thing as the end behavior of a function.

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Pretend that's some quadratic.

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The limit as x approaches infinity of x squared is infinity.

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So the limit as x goes to infinity, we want to know as x gets larger, what is f of x

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doing?

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And for quadratics, what's it doing?

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Getting larger and larger and larger.

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All polynomials either go up to positive infinity or go down to negative infinity.

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So our answer here would be infinity.

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Let's say some fourth degree polynomial.

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as x goes to infinity of whatever this function is.

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Again, if it's a polynomial, the answer is either going to be positive or negative

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infinity.

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So here, as x gets larger, this keeps going down and down and down and down.

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So this would be negative infinity.

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Okay.

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Oops.

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Let me do this one.

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Rational functions often had horizontal asymptotes.

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Do we remember this?

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And here, the limit, let's call this g of x, as x goes to infinity, this gets closer

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and closer to the number 3.

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It will never be 3, but this has a horizontal asymptote, the answer is 3.

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And then there's cases like sine of x.

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Let's say infinity or negative infinity.

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This is bounded, but will ever go approach a single number?

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No, so like this, something like this, the answer

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does not exist. So all the answers to limit as X goes either positive or negative

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infinity

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will either be positive infinity if it keeps going up forever, negative infinity if

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it

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examples. What kind of function is 4 over x? Rational.

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OK. You can kind of do some math with infinity.

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It's kind of like direct substitution.

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OK. 4 divided by a very, very large number is?

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Yeah. Well, yeah, I won't get there.

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Sorry. Let's say this is some number, very large number.

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4 divided by a very, very large number is a very, very small number.

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okay and the larger you plug in the smaller gets okay so in terms of limits

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The larger the magnitude of x becomes, the smaller the quotient becomes.

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zero and the same is X goes to negative infinity any number over negative

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infinity keeps getting smaller and smaller and smaller even though it's negative we

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don't care that is also zero. So when you try the direct substitution even when

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we're plugging in infinity anytime you have any number over infinity the limit

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It always goes to zero.

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Let me change this to minus.

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Sorry, let me change this one to minus.

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Here we got another rational function.

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If you try plugging in infinity, we get 3 times infinity squared minus infinity plus

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2, or 4 times infinity squared minus 5.

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What is infinity squared?

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3. Infinity times 3 is infinity. Infinity minus infinity plus some number 2. What

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are

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we doing here is a good question. You can still think of negative infinity plus 2 as

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negative infinity either way what is infinity minus infinity no we don't know

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that's an indeterminate form very important infinity minus infinity is an

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indeterminate form infinity plus infinity is not indeterminate what does

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that equal that's just infinity a large number plus a large number is a large

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But a large number minus another large number, we can't really do math with

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infinity.

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So that's an indeterminate form.

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So our point is we need to do something else here.

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And what are we going to do here?

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Some of you were probably taught last year just erase this stuff.

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Can we just erase stuff?

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No.

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The answer is sure and poor but why?

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Because Mr. Williams just said erase it.

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Erase it.

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I'm going to use the last 3.

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Oh, that was canceled.

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The x goes to infinity of 3 fourths. It's just 3 fourths.

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Okay, we just can't erase stuff.

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No, we can't.

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So what can we do?

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How can I show what's going to be equal to 3 fourths?

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You might be able to factor, I don't know if that's going to help.

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I don't even know if these factor.

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Yeah, I guess it does.

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That's not going to help.

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There's a trick.

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We should add it to our tricks.

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Let me copy this real quick.

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We're going to multiply the numerator and denominator by something.

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I'll do that.

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What?

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Which is just the number one,

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as long as X is not zero and it's definitely not zero.

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We're going to infinity large numbers.

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And I'm gonna choose the smaller of the two degrees.

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This is a second degree in the numerator,

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a second degree, they're the same degree.

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I'm going to choose 1 over x to the smaller degree, but they're the same.

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And then I distribute that.

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After multiplying by one over x squared, the expression becomes three minus one over

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x plus two over x squared, divided by four minus five over x squared.

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That over x plus 2 over x squared.

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OK.

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And then using our limit laws, the limit of a quotient is the quotient of limits,

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and

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then the limit of a sum is the sum of limits.

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We're not going to write all that, because it's just writing lots of different

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limits.

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But, let's just look at each single piece.

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As x goes to infinity, what does 1 over x go to?

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Zero.

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Two over infinity squared is two over infinity, which goes to zero.

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Five over infinity squared goes to zero.

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So the numerator goes to 3 minus 0 plus 0 and 4 minus 0, the answer is 3 fourths.

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When the degrees are equal, the limit is the ratio of the leading coefficients.

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taught right we look at the two degrees they're the same it's just the leading

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coefficient divided by the leading coefficient but this is why it would

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anything change if I went to negative infinity? No, the negative infinity that

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still goes to 0 0 0 we still get 3 4ths. Okay so this rational function has a

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horizontal asymptote I don't know what it looks like in between but definitely

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a 3 4ths goes here and here or something and it has some I don't know what it looks

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like in

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but something like that

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So here's another rational function.

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Right, this is a fifth degree polynomial as x goes to positive infinity, this goes

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to

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positive infinity, this is a third degree polynomial with a negative leading

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coefficient.

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It goes to negative infinity.

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And what is infinity divided by negative infinity?

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Indeterminate form.

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Don't know.

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OK.

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So, we've got to do some more stuff.

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So what are we going to multiply by?

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One over x cubed divided by one over x cubed.

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I'll show you if you don't choose the smaller of the two what can happen but we

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distribute

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As X goes to infinity, 5 over infinity goes to zero, 1 over infinity cubed goes to

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zero,

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negative seven over infinity cube goes to zero.

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Okay.

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What does infinity square go to?

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Over negative three.

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What is infinity divided by negative three?

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Negative infinity.

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A positive number, right?

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Infinity stands for a very, very large positive number

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divided by negative three

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is a negative some very large number.

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So it's a negative, negative, negative.

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Okay, I chose the smaller of the two degrees.

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If you didn't, you run into problems.

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Let me show you what happens if

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Let's say I chose x to the fifth.

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You would get 1 minus 5 over x cubed plus 1 over x cubed over negative 3 over x

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squared

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minus 7 over x cubed.

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You don't have to write this down.

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So x goes to infinity, this goes to zero, this goes to zero, this goes to zero, this

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goes to zero, and we get one minus zero, plus zero over zero, plus zero, one over

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zero.

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And what does a number over zero go to?

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In terms of limits.

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Either positive or negative infinity.

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But what we don't know here, since we did it here, we don't know, right, the

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denominator goes to zero.

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But we don't know if it's from the left side, which is a negative number, or from

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the right, it's a positive number.

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So we don't know what this ends up as positive or negative infinity.

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OK, so we don't want to do this. OK, so anyways, the point was,

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do one over the smaller of the two degrees and then it will show you whether it's

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positive or negative infinity.

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Let's try another one.

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Again, if you plug in infinity, you get infinity over infinity indeterminate form.

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So what am I going to multiply by?

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The numerator has degree one and the denominator has degree two, so multiply by one

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over x.

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So one over x, one over x.

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A smaller two degrees.

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this goes to 0, this goes to 0

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1 plus 0

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this goes to negative infinity

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minus 0

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what is 1 over negative infinity go to?

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any number over positive infinity goes to 0

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So these are three different cases.

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One here, when the degree in the denominator is greater, this always ends up going

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zero,

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but this is kind of why we showed it.

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Here when the degree in the numerator is larger, it's either going to go to positive

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or negative

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infinity.

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You have to use this to figure out which one and then when the degrees are the same

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It is always the leading coefficient divided by the leading coefficient

202
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And this was why it's the case

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Okay, so that was rational functions

204
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What type of function is this?

205
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Exponential.

206
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Okay, you can kind of do infinity math of this

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Three times infinity

208
00:21:18,410 --> 00:21:23,250
Three times infinity is still infinity. What is e to a very very large number?

209
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Like what's e to a thousand at which is not very large

210
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It's an enormous number.

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Big, big, big, big number.

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Okay, this is infinity.

213
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And then, let's see what happens as we go to negative infinity.

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What is e to a negative exponent?

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It is 1 over e to positive infinity.

216
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Okay, e to positive infinity is infinity.

217
00:22:42,620 --> 00:22:44,300
What is 1 over infinity?

218
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Zero.

219
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It goes to zero.

220
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Okay.

221
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All exponential functions are either the 3x,

222
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looks something like this.

223
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All exponential functions on one side,

224
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it goes to either positive or negative infinity.

225
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On the other side, we always have a horizontal asymptote

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of zero.

227
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So that's the end behavior for all exponential functions.

228
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If this were like negative 3x, it would just be flipped.

229
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this will go to 0 and this would go to infinity. So you do got to be careful to sign

230
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this up.

231
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And then we'll do the other case.

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For the radical expression, first evaluate the negative-infinity case.

233
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Nine times negative infinity squared plus negative infinity, well forget that.

234
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This is a quadratic.

235
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Where does a quadratic go as you go to negative infinity?

236
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It goes to positive infinity, right?

237
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That quadratic looks something like this.

238
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So the inside here goes to positive infinity.

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The square root of positive infinity is... square root of a very big number is still

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a really big number.

241
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So that goes to infinity.

242
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And then we got minus 3 times negative infinity.

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What does negative 3 times negative infinity go to?

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Infinity.

245
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And so our answer is infinity.

246
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This is infinity.

247
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Now the other one is much more interesting.

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Here if you plug in infinity, we get infinity minus infinity, which is what?

249
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In determinant, in terms of limits it's an indeterminate form.

250
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Ok so what are we going to do here?

251
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It's one of our old tricks.

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We're going to multiply by the conjugate, thank you whoever said that.

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Okay, so we're going to go the square root of 9x squared plus x plus 3x.

254
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plus x. The 2 milli terms cancel out, we get minus 9x squared. And now what happens?

255
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These

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cancel.

257
00:26:24,290 --> 00:26:27,330
x over x squared.

258
00:26:33,970 --> 00:26:37,410
Which we plug infinity, we still get infinity over infinity.

259
00:26:46,910 --> 00:26:52,591
We're either going to multiply by 1 over x, or we're going to factor out an x out of

260
00:26:52,591 --> 00:26:53,341
here.

261
00:26:53,930 --> 00:26:56,430
Let's do this one.

262
00:27:04,370 --> 00:27:07,730
We're running out of time, so I've got to speed this up.

263
00:27:19,390 --> 00:27:22,011
After rationalizing, divide by x and move the positive x into the square root as the

264
00:27:22,011 --> 00:27:22,830
square root of x squared.

265
00:27:26,310 --> 00:27:29,290
Is x equal to the square root of x squared?

266
00:27:34,870 --> 00:27:37,010
This is very important.

267
00:27:38,990 --> 00:27:42,357
The identity x equals the square root of x squared is valid here because x

268
00:27:42,357 --> 00:27:43,107
approaches positive infinity.

269
00:27:46,990 --> 00:27:50,030
If x is a negative number this is not true.

270
00:27:50,770 --> 00:27:58,930
In fact, if x is a negative number, we have to put a negative here.

271
00:28:02,230 --> 00:28:03,130
Oh, it's about to ring.

272
00:28:03,350 --> 00:28:04,250
So which one is it?

273
00:28:04,210 --> 00:28:05,330
Is it a positive or a negative?

274
00:28:05,410 --> 00:28:06,790
Which one am I going to substitute in?

275
00:28:08,210 --> 00:28:13,269
Since we're going to positive infinity, I'm going to substitute this x right here

276
00:28:13,269 --> 00:28:14,019
for

277
00:28:13,630 --> 00:28:20,250
this, we get the limit as x goes to infinity.

278
00:28:22,250 --> 00:28:28,290
Get rad nine x squared plus x over rad x squared,

279
00:28:30,430 --> 00:28:33,830
which now I can make one big square root.

280
00:28:41,170 --> 00:28:50,730
We get the square root of 9 plus 1 over x plus 3.

281
00:28:51,410 --> 00:28:52,730
Hold on one sec.

282
00:28:54,830 --> 00:28:58,910
Now as x goes to infinity this little guy goes to zero.

283
00:28:58,910 --> 00:29:03,470
As x approaches infinity, one over x tends to zero, so the limit is one sixth.

284
00:29:05,470 --> 00:29:08,370
Okay, be real careful when you bring in

285
00:29:09,070 --> 00:29:10,510
into square roots.

286
00:29:11,470 --> 00:29:13,010
Sometimes this, sometimes that.

287
00:29:13,950 --> 00:29:14,850
Okay, the homework is some delta math.