13. When is a polynomial f(x) of at most the order of a polyno...

Question

13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let fx equal x 6 minus x and g x is equal to 2 multiplied by x 6 + 7 is f x at most the order of g of x as x approaches infinity. OK, so it appears for this particular problem we're asked to consider the following statement, and our statement once again is we need to let F of X be equal to x 6 minus x and g x is equal to 2 multiplied by x 6 + 7. And we're asked to determine whether it will be a yes or B no, that F of X is at most the order of g of x as x approaches infinity based on the information that is provided to us for the prom itself. So will our final answer be a yes or B no? So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to state and note what our definition is. So we need to let F of X and G of X be positive. For sufficiently large values of x, and we need to note that f of x is equal to o g x, and this will mean that there is a positive integer M such that fx divided by g x is greater than. Or so once again, F X divided by GX will be less than or equal to M. As X approaches positive infinity. So that's what FX is equal to O G X means. It means that there's a positive integer M such that FX divided by GX will be less than or equal to M as X approaches infinity. And if these conditions hold true, we can say that F of X is of at most the order of G of X as X approaches infinity. So with that in mind, our second step is we need to form the ratio, and we will need to observe what the behavior will be as x approaches infinity. So we need to write that fx divided by g x will be equal to x 6 minus x divided by 2 multiplied by x 6 + 7. Which will be equal to x 6 multiplied by parentheses of 1 minus x divided by x 6 divided by x 6 multiplied by parentheses of 2 + 7 divided by x to the power of 6. And we need to note that as x approaches positive infinity, we will be able to write that the limit as x approaches infinity of fx divided by g x will be equal to the limit as x approaches infinity of x 6 multiplied by 1 minus x. So no, so it's gonna be x 6 multiplied by 1 minus x divided by x 6. Divided by x to the power of 6 multiplied by parentheses of 2 + 7 divided by x to the power of 6, which will be equal to the limit as x approaches infinity of 1 minus 1 divided by x to the power of 5. Divided by 2 + 7 divided by x to the power of 6, which will be all equal to 1/2 or 1 divided by 2. So for our 3rd and final step, we need to make our final conclusions. So based on our knowledge that, based on our prior knowledge of what we've learned together so far, we can safely conclude. So as X approaches positive infinity. That will mean that as a result f of x divided by g of x will be less than or equal to 1/2. So as a result, that will mean that yes, so that's our final answer is yes, yes, that will mean that fx is of at most the order of g of x as x approaches infinity. So looking at our multiple choice answers, our final answer, once again, without a doubt, will be a yes. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.

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