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23 (2 bonus Points) Lel [n be the number of parlilions of which have exaclly two parls of size Lel L(x) Clnx' be the generaling funclion for (ln } . (Let lo an...

Question

23 (2 bonus Points) Lel [n be the number of parlilions of which have exaclly two parls of size Lel L(x) Clnx' be the generaling funclion for (ln } . (Let lo and [1 Give an argumenl using only generaling functions for why for n > 3,Pn

23 (2 bonus Points) Lel [n be the number of parlilions of which have exaclly two parls of size Lel L(x) Clnx' be the generaling funclion for (ln } . (Let lo and [1 Give an argumenl using only generaling functions for why for n > 3, Pn



Answers

In Exercises $13-18,$ investigate the behavior of the finction as n or $x$ grows large by mating a table of function values and plotting a graph (see Example 4). Describe the behavior in words.

$$
f(n)=\left(1+\frac{1}{n}\right)^{n^{2}}
$$

Okay, so for this question, if we said P is equal to the limit As end approaches infinity off base of N plus one, but it by a ace up and we end up with ex cute X cube multiplied by Elena and lets one bye bye l end. And this is equal to the limit as an approaches infinity, using local tolls rule off X cubed, multiplied by one over and plus one provided by one over end simplified down to end over and plus one. This is just equal to the upside value of execute so that we can say that p is less than when when the absolute value of X cubed is less than one. Therefore, we consider the Siri's convergence absolute be on the interval from negative 121 for the end point of X is equal to negative one. The Siri's becomes one over l and off and which is a divergence. See respect the compares in test because one over Ellen of end is greater than one over and one of her and is a divergent Herman. Siri's for X is equal to negative one. This is Rex. People put one x equals negative one. We end up with negative one to the power over n divided Bye, Ella of n, which is a convergence Siri's by the likeness tests. Therefore, we know that this serious converges for values of X between native born and on one and I.

It is given that for the number of that noticeable for Lincoln and Sides function is in the form for polynomial Poppins. Quit it minus three. By doing, we need to give the simplified expression by taking out de boned. Better half. So everybody had to divide by half isn't were toe three by two into two. This is a working three. So the temblor is my normal expression becomes our former and square minus 30. Yet I can take any woman I get in my best three as the oil answer.

To see which of these functions grows the slowest. Let's first compare log base to events squared and end. We'll do this by looking at the limit of the quotient so we can use a change of based formula to write log based you livin as Elena them over Alan. Swear Eleanor too squared and we saw the squared left over. Now we can pull out one over Ellen of two squared, and so we're looking at the limit is then approaches Infinity of the natural log of end squared Times said We can use, like, potential here to evaluate this limit. So we get one over Alan of two squared times. The limit is end approaches infinity of two times ln of end, Times one over end. Now this one over end comes from using the chain Well, so that's the derivative of the natural log of M. And the derivative of the denominator is just one. So, really, we're looking at two over Alan of two squared times the limit of Elena ven over end. Once again, we'll use lobby Taj rule to evaluate this limit so you get to over the natural log of two squared times the limit is then approaches infinity of one over and over one. So since the limit, as I'm a purchase infinity of one of her and zero we see that love. These two of end squared grows at a slower rate land. And so now if we compare log based who have n squared with the swearing event Times Law based to a Ben we need to look at the limit is that approaches infinity so we can cancel out one of the log based use of end to get the limit. Is an a purchase and finishing of long place to have end over the squared event. What scan will do a change? A base formula. So the numerator becomes ln event over. Eleanor, too, and doing a little bit more rearranging. We get one over Ellen of two times the limit. Is that a purchase? Infinity of Ellen of them over the square event here. Will you sabotage role to a fight with the limit? So we get one over Ellen of two times the limit is that approaches infinity of one of her end in the numerator, and the denominator becomes 1/2 times and the three house being in the sub again, we get one over Eleanor two times the limit is in a purchase. Infinity. Ah, one over 1/2 times and to the 1/2. Now it's an approaches infinity. Here this limit is zero, which means that lobby's two of and squared grows at a slower rate than the square root of n times. Love based two of end. So this is this to the conclusion that long face two of n squared is actually the function the girls at the slowest rate, which means that the algorithm that takes the order of that many steps is actually the most efficient, because in the long run it'll grow slower than the other two objects Now Partney asked you to graph thes three functions. So the red graph here is n the green one is the square root of n times law based two of end and the blue one is long waist who have n square

Okay. In this problem, it is required to find and terrible off conversions for these Siris. We will start by taking a off n equally cool toe factorial over in a factory 43 off the Playboy Ex boat and so limit intends to infinity for he's up in plus one over. He's up in with the equal to toe plus two factorial over and plus one for custodian Old Porcari X and plus one deploy to end Fuck studio eggs sport in Hoover in factorial Oh, no. Or three. We can simplify by removing exporter in with export in and in factorial course three who is factory upholstery and to end factorial ways with these factory. So our rhythm we're to be to end close to deploy boy to end plus one This is the remaining off to end. Plus two factories must deploy Boy of Solu tex over and plus one old or three or three. If you'd like to simplify more, we can. We'll deploy the presence so we will have four and forced to plus six yen lost too over and post to be plus three And yours too close the end plus one. Of course from the playboy up. So you, Tex, if we divide by one for imports three, four months to Playboy won over imports Torreon. Those cases deploy one over and course we deploy boy on over on course three. So we will have for over in close six over and it's queer both toe over. Name course. Three over one bluff. Be over in plus three over in the square, plus one over and Porcari Deploy Boy Factory Alexe. All these values equals toe one because anything over infinity equal to zero story. So zero plus zero plus zero because the zero over anything is zero mark deployed by anything is zero. So our result will be equal toe zero. That means that or equal toe affinity. And this year's is conversion. Dhe we any value affects


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