#Calculate the limit

121 messages · Page 1 of 1 (latest)

desert rain
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The b) one

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copper ospreyBOT
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round wasp
desert rain
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But it still looks weird

round wasp
desert rain
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I meant

desert rain
desert rain
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To get rid of the roots

round wasp
desert rain
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Well uhh

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Someone's told me thatt

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Wait

round wasp
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Okay, no.

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"Someone told you that" is not how we know things in math.

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We prove things in math.

desert rain
round wasp
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So you know what the radical conjugate is for a square root is, right?

round wasp
desert rain
desert rain
round wasp
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So we know that (a - sqrt(b))(a + sqrt(b)) = a^2 - b, right?

desert rain
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Yupp

round wasp
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But let's let c = sqrt(b), then it becomes (a - c)(a + c) = a^2 - c^2. It was the difference of squares formula in disguise all along.

desert rain
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Yaa I kinda saw that I think hahah

round wasp
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So in order to find "the conjugate" of a cube root, we need to use the difference of cubes formula.

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Do you know it?

desert rain
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The a³-b³ one?

round wasp
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Yes.

desert rain
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I think so

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It goessss

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(a-b)(a²+ab+b²)?

round wasp
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No, I'm afraid.

desert rain
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Ah no the other way around

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Let me edit

round wasp
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That's a^3 + b^3.

desert rain
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There

desert rain
round wasp
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To be fair, it doesn't overly matter in this context. The important point is that every uncubed factor of a gets canceled.

round wasp
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Rotor, I have it well in hand.

desert rain
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a = cbrt(1-3x) and b = -2

wicked skiff
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Ok sorry

round wasp
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Hence, the "conjugate"?

desert rain
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It'll beee

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(Cbrt(1-3x) - 2)((1-3x)^2/3 + 2cbrt(1-3x) + 4)

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I thinkk

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Sorry hard to type that on phone😭

round wasp
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...no. Remember, we just established that it's a^2 + ab + b^2.

round wasp
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Because that's what we have to multiply and divide by.

round wasp
desert rain
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Ohh

desert rain
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The longer one

round wasp
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Are you writing this down? On paper?

desert rain
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Not really

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Should I

round wasp
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Yes, absolutely.

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Always do math on paper.

desert rain
round wasp
desert rain
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Okayy

round wasp
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So begin from the beginning, and then write down everything we noted.

desert rain
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Okayy

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Noted

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But I'm not sure what the conjugate will be

round wasp
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What did we say it would be?

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What was the difference of cubes formula?

desert rain
round wasp
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Get it down on the paper before you tell me.

desert rain
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I've written it down already

round wasp
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Okay, good.

desert rain
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Why are we using the formula for the difference of cubes though if we're adding them

round wasp
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So now, again putting it on the paper before answering, what's a and b?

desert rain
round wasp
desert rain
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-2

round wasp
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Right.

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Therefore, the "conjugate" is what?

desert rain
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Let me write it down first

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(1-3x)^(2/3) - 2(1-3x)^(1/3) + 4

round wasp
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Right. Therefore we multiply top and bottom by that, and on top we get what?

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We already know what we get, so you don't have to calculate it by hand.

desert rain
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Yaa the root disappears

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And the 2 gets cubed I think

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So we get

round wasp
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And that's why we want the difference of cubes.

desert rain
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1 - 3x + 8?

round wasp
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It's not the "difference" part we care about, it's the "of cubes".

desert rain
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Oh okayy

round wasp
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We do this so that we cube the cube root and don't get any weird fractional powers along the way.

round wasp
desert rain
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Alrrr

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Let me write

round wasp
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No!

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It should be -3x.

desert rain
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Oh yesyes 😭

round wasp
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And so what does that simplify to?

desert rain
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-3(x-3)

round wasp
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Bit more simplified than I was asking for, but then, again getting it down on the paper first, what does that mean our whole expression simplifies to?

desert rain
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Yes I'm writing as we're speakingg

desert rain
round wasp
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Right.

desert rain
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And we're left with -3 in the numerator and all that what we multiplied by in the denominator

round wasp
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Which is what?

desert rain
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Let me calculate

round wasp
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No.

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I mean just tell me what's in the denominator.

desert rain
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Ohh

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(x-3)(1-3)^2/3 - 2(1-3x)^1/3 + 4

desert rain
round wasp
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Cool.

desert rain
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I think that's allll

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I don't like these conjugations too much😭

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Tysmmm tho you helped sm

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+close

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# desert rain +close
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