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Test: Test 1 (Chapt.2 Test) This Question: ptCalculate the following limit using the factorization formulaa" = (x-a)(xn-xn ~ Za+xn-3a2xa" -2 +an-1) where ...

Question

Test: Test 1 (Chapt.2 Test) This Question: ptCalculate the following limit using the factorization formulaa" = (x-a)(xn-xn ~ Za+xn-3a2xa" -2 +an-1) where n is a positive integer and a is a real number: x5 243 lim X-3 X-3x5 243 lim X-3 X-3 (Simplify your answer: )

Test: Test 1 (Chapt.2 Test) This Question: pt Calculate the following limit using the factorization formula a" = (x-a)(xn- xn ~ Za+xn-3a2 xa" -2 +an-1) where n is a positive integer and a is a real number: x5 243 lim X-3 X-3 x5 243 lim X-3 X-3 (Simplify your answer: )



Answers

Calculate the following limits using the factorization formula $x^{n}-a^{n}=(x-a)\left(x^{n-1}+a x^{n-2}+a^{2} x^{n-3}+\cdots+a^{n-2} x+a^{n-1}\right)$ where $n$ is a positive integer and a is a real number. $$\lim _{x \rightarrow 2} \frac{x^{5}-32}{x-2}$$

Okay, So would start the factory or your reading. Noted Teoh Studio, about five. And now, using our factor irrigation for me, we get X minus two times Sex, Andreas once that's four. And then we have plus except factory. And then times aim. So a in this case is too. And then we have us. It's about you and then a squared. That's for men. Plus, um, X times is your part of three start ups, and then we have plus a to the power of for which is 16. And this is all over X minus two. We can now, uh, catch out or exercise to know that cannot be able to to or we have the vintage in their own Now using drugs sellable we got treated are four plus to tip our three times two plus two. Do not two times, for that's two times age and 16. I'll put your feet are calculated. Okay, so we have two times 2 to 3, and then we have four terms to depart to two times eight and plus 16. That's equal to Eddie

We're us Define the amendment as X approaches a of extra benefit minus eight to the fifth over, X minus A. As always, your first asked ourselves there anything that's stopping us from simply putting are a There is. This is the nominative here we that anomie or to not people zero we plug in a nominator will equal zero. So we need to find a way to simplify this expression. The way that I think about this is that we could factor at the top here. But it might seem, Don't do that first. Luckily, the problem gives us a formula that we can use defector outs, the top here, the normal it is we have ecstasy and minus eight to the end it factor this into X minus A that multiplies by X to the n minus one Uh, plus a times X to the n minus to plus a square times x to the end of minus three dot plus dot dot Down plus eight to the end, minus two times Tax uh, Times X plus A to the end minus one. So there's a method to this madness. Right here. We see that X to the n minus one. We see that for this monstrosity term right here Each time that X each term X goes down on power, whereas for each term A goes up by a power. So we have 80 here and the first a square, its third Yana. And eventually we will get a to the N minus one. Just our biggest exploited that we have in this truck. So using this model right here, we've been appliance X to the fifth, minus eight to the fit and in this case, will people five. So that makes our life pretty easy. Here we have X minus A as usual, and an artist who cancel out to the top a term for the top of the bottom. Let's continue fighting this out. And here we'll get exit four plus eight times X to the third, plus a square X to these X square, right? Yes, yes, yes, that's true. Plus a to the third Times X plus A to the fourth. Right now we can play this back into original problem, so we'll have the minute as X approaches a of X minus a times. This whole party right here. It's the fourth plus a X to the third, plus a squared X squared, plus a huge X plus A to the fourth. All of that over X minus A that cancel out the exercise. A. They are simply left with the limit as X approaches a of exit fourth plus eight times X to the third, plus a squared X two X Square plus a cube fax plus a to the fourth. Now nothing stopping us from playing in our A here and who do this. We'll get age of the fourth over here, plus eight times eight of the third, which is just a day before over here. Plus another a squared plus a square, a square Times Square, which is hated fourth plus another eight to the fourth and, of course, the last stage of the fourth. So ultimately, this will equal fine a to the fore. Boxer answer and nice green check Mark

We are asked to solve for this limit here. Now, the first question actually should always ask ourselves, Is there anything stopping us from plugging a one? It turns out there is. There's distraction bar right here. We need to ensure that whatever is that, the denominator does not. Zero. Unfortunately, we're playing our ones become one minus one, which would be zero, and we would be divided by that zero, which would be illegal. So we need to think of some way to simplify this top expression here so that we can cancel something out, maybe with the bottom, or manipulate this expression somehow so that we can get rid of this annoying X minus one on the bottom here. Well, it might seem daunting to factor this top part first, but noticed that the problem actually does give us a nice formula that we can use. Let's write down this formula right here. So it's X to the n power minus a to the power you're giving something like this, you can always fact ripped into X minus a times X to the n minus one plus a to the X to the n minus two plus a square time tax to the ends of minus three plus dot, dot, dot plus a to the end, minus two x plus. Uh, eh to the end, minus one. And if you notice here, there's a pattern to this monstrosity. There's a pattern to the madness R sorry. There's a method to the madness. That's how the saying goes. Yeah. So for each power that X decreases. As you can see, XT is decreasing by one power Here A is increasing by one powers of here raise that has power zero a power one to happen. Eventually a becomes the largest exponents camp here, which is n minus one and swaps places with the ex. So, using this information that supply into this problem, we can think of X to the six minus one as X to the sixth, minus one to the six. If this is the case, we can plug it into here and we will get X minus a logistically X minus one times And here we are receiving, we can cancel out the top and bottom, but let's finish the rest of it. First X to the N minus one and equal to six years of this becomes X to fit. That was because a times aspects of the fourth plus a square I'm sorry exactly what I'm going to write Down a plus one one times X to the fourth plus one square times next to the third plus one Hugh times X square plus one to the fourth times X plus one to the fit close parentheses. So we simplify and take out all these ones. Here this becomes X the sixth minus one to the sixth equals X minus one times X to the fifth plus extra before plus X to the third plus x square plus X and plus one the end grace. Now that we have this, let's put this all over X minus one. So if we plug this in for our top here, you will get limit Has x a purchase? One of this whole thing. So we have explain this one times X to the bit plus X to the fore plus X to the third plus x squared. My hand is the entire plus X plus one finally all over X minus one. Now we cancel out the excellent It's one in the excellence one here. Great finally get those out of the way and this becomes limit as okay, we have to read it again. Don't me living as Ice approaches one of X to the fifth plus X to the fore, plus next to the third plus X squared plus X plus one. Now there's nothing Something is from cooking in our one. So let's plug it in. I don't want to write it all out. So that went to the fifth becomes one plus one of the four it becomes one plus one to third becomes one plus one. Squared equals more plus one plus one. And this equals six. Right one, Teoh six. Yet people six. That is our answer. Let's put a nice screen check more. We're finally done with this question.

This question is asking us to find limit as X approaches a of extra fifth minus eight of the fifth over X minus A, and it asks us to use a fact factories ation formula, which is seen here. And to do this typically, what you want to do is you want to find the portion of the limit. In this case, it would be extra the fifth minus eight of the fifth. And then you want to put it into this form right here. And do we know what this limit? We find that the limits as X approaches a is equal to X minus a and then you start throwing the rest of this function this formula and and starting with your the X minus one would be X to the fourth, and then the plus X n minus two will be extra. The third time's a, as you can see here and then do, plus X n minus three, which would be X squared times a a squared and that could be seen here. And then you go to the next value, which would be X times a in minus two to be a cube, as you can see in this part and then simply a in minus one, which would be eight of the fourth, as you can see here at the end of this one and all of that, of course, is divided by the denominator is seen here X minus A. A real convenient thing about the factory's ation formulas. You almost always have the ability to cross out this first time, and typically you want toe. You want to find the value of that limit that will allow you to do so so this will become limit. X approaches a of X to the fourth plus X cubed a plus X squared, a squared plus X, a cubed plus hate to the fourth and then plugging in your value of a for X. You get the limit as X approaches. A is equal to eight of the fourth plus eight of the fourth when you play a into this X cubed because a cube times A which is aided the fourth plus eight of the fourth again because you have a square times a a squared, which is simply a to the fourth. Then you have a times a cute, which is also ate of the fourth, plus that last value dated before, and this becomes equivalent to 58 of the fourth. So the limit as X approaches a of extra. The fifth minus eight of the fifth, divided by X minus A is simply 58 to the fourth.


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