111. True, or false? Give reasons for your answers.
c. x...

Question

111. True, or false? Give reasons for your answers.

c. x = o(x + ln(x))

Accepted Answer

Welcome back everyone. Determine whether the statement is true or false. X2 equals little O of X2 plus LN X. A says true, and B says false. We can begin this problem by remembering the definition of little O. If we have F of X equals little O of G of X, it means that as X approaches infinity. The limit. Of F of X divided by G of X is equal to 0. In practice, it means that the G of X grows faster than F of X. So let's check if this is true. We're going to evaluate the limit as X approaches infinity of X2 is our F of X and G of X is X2. Plus Alan of X. To find this limit, what we can do is simply divide all of our terms by x2d, which gives us limit. As x approaches infinity of 1. Divided by 1 plus LN of x divided by x squared. So now let's analyze each term. In the numerator, we have a constant which is one. In the denominator we have one less. Now, as X approaches infinity, a land of X is going to approach infinity, right? And X2d is going to approach infinity as well, right? Because we have infinity squared and that's infinity. However, let's remember that polynomials grow faster than natural logarithms, so that this indeterminate form. is actually equal to 0 because our denominator, which is a polynomial function, grows faster than the numerator, which is a logarithm. So we can essentially say that LN of X divided by X2d is going to approach 0 as X approaches infinity, and calculate the limit as 1 divided by 1, which is 1. Now because this is not equal to 0, we can conclude that the statement is false and the answer is option B. Thank you for watching.

111. True, or false? Give reasons for your answers.c. x = o(x + ln(x))

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111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))