9. True, or false? As x→∞,
c. x = O(x+5)

Question

9. True, or false? As x→∞,

c. x = O(x+5)

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. Decide whether the equality is true or false as x approaches positive infinity. X2 + 3 X is equal to O of X 3. OK, so we're asked to decide whether or not the equality will be true or false as x approaches positive infinity, given our expression, and we're asked to determine whether or not our final answer will be a true or B false. So once again we're deciding whether or not this equality is true or false as x approaches positive infinity. OK. So, with that said, our first step that we need to take is we need to state the definition. So we need to note that F of X will be equal to O G X, as this will be as X approaches positive infinity, which means there exists constants where capital M is greater than 0. And that x subscript 0, such that the absolute value of F of X is less than or equal to M multiplied by the absolute value of G of X. Fantastic, and this will be for all values of x is greater than or equal to x subscript 0. So once again, this is for all x is greater than or equal to x subscript 0. So our second step is we need to identify the functions, meaning we need to let F of X be equal to X2 + 3 X, and we need to state that G of X will be equal to X 3. Fantastic. So, our third step now is we need to note that for x is greater than or equal to 1, we need to note that x 2 is less than or equal to x 3, and we need to note that 3 x is less than or equal to 3 multiplied by x 3 when we multiply by x is greater than or equal to 1. OK. So, for our fourth step is we need to combine our inequalities. So we need to note that X2 + 3 X is less than or equal to X 3 + 3 x 3, which is equal to 4 multiplied by x 3. And this is for X's greater than or equal to 1 by addition. So I put a little positive symbol inside quotation marks. So With that said, our fifth step now is we need to conclude our constants. So we need to choose that M is equal to 4, and that X subscript 0 will be equal to 1. So that will mean that the absolute value of X2 + 3 X. Which once again, anything inside of absolute value symbols will become a positive value. So the absolute value of X2 + 3 X is less than or equal to M multiplied by the absolute value of X 3. And this will be 4 X is greater than or equal to X subscript 0. So, step number 6, which is our final step, is we need to state our final result. So therefore, X2 + 3 X will be equal to. Oh, so once again, X2 + 3 X will be equal to O of X 3. As X approaches positive infinity. And this will mean that because this is the case, that will mean that our final answer will be true. So the statement without a doubt is true. So our final answer, once again, looking at our multiple choice answers will be A, true. And that's it, we've solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

9. True, or false? As x→∞,c. x = O(x+5)

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9. True, or false? As x→∞,
c. x = O(x+5)