11. Show that if positive functions f(x) and g(x) grow at the ...

Question

11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).

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 p x and q x be positive for large x and suppose the limit as x approaches infinity of p x divided by q x is equal to alpha, with 0 is less than alpha, alpha is less than infinity, which answer is correct? So it appears for this particular problem we're to determine which of our multiple choice answers is correct, considering that if we let p of x and q x be positive for large values of x. And we suppose that we're given a limit as x approaches infinity of p x divided by q x is equal to alpha, and we're given an interval of 0 is less than alpha and alpha is less than infinity, we're asked to determine once again which of our multiple choice answers is correct based on the information that is provided to us by the prom itself. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer or answers may be. A is P X is equal to O Q X, and QX is equal to O P X. B is QX cannot equal O P X. C is PFX cannot equal O of q x. D is Q of X is. Cannot equal, so QX cannot equal O P X, and P X cannot equal O Q X. OK. So with that in mind, our first step that we need to take is we need to consider the definition of the O notation, meaning if F of X and GX are positive for large values of X, then the function F is of at most the order of the function G. As X approaches infinity. And if there is a positive integer such as L, such that FX divided by G X is equal to L, and this will be for large values of X. So, with that in mind, for our second step, we now need to check if P X is equal to O Q X. OK, so here we will be able to see based on what the problem tells us that PX and QX are positive for large values of X. And that once again our limit as X approaches infinity of P X divided by Q X is equal to alpha, with 0 is less than alpha and alpha is less than infinity, and these do match the conditions for P X is equal to O Q X. For our third step, we need to check if Q X is equal to OP X. So now, P X and QX are positive for large values of X, and the limit as X approaches infinity of PX divided by Q X is equal to alpha, which will mean that the limit as x approaches infinity of Q X divided by P. Of X is equal to 1 divided by alpha, which will be equal to beta. OK. And we need to note that with, so we have an interval with 0 is less than alpha and alpha is less than infinity, that will mean that 0 is less than 1 divided by alpha, which is less than infinity, which will mean that 0 is less than beta, and beta is less than infinity. And these also satisfy the conditions for Q X is equal to O P X. So for our fourth and final step is we need to consider and note that both P of X is equal to O Q X. And so once again, let's make a note that both, let's make a note that both PX and Q X is equal to O P X are true. So let's make a note that they're both true. So that will mean since both of our. Answers for PX and Q X are true. That will mean looking at our multiple choice answers. Our correct and final answer without a doubt will be multiple choice answer letter A, which once again is P X is. Equal to O of Q X and Q X is equal to O P X. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully it helped, and I can't wait to see you in the next video. Bye.

11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).

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