3.2 Asymptotics

Which of the following is true about the little oh and Big Oh?

Exercise 1

 If \(f = o(g) \), then \(f = O(g) \). 

 If \(f = o(g) \), then \(g \neq O(f) \). 

 When \(f \sim g \) and \(f \neq o(g) \), \(f = O(g) \). 

 When \(f \nsim g \) and \(f = o(g) \), \(f = O(g) \). 

 If \(f = O(g) \), then \(f = o(g) \). 

 

Which of the following is NOT correct about the little oh?

Exercise 2

 \(x^{a} = o(x^{b}) \) for all integers \(a \) and \(b \) 

 \(b \cdot x^{a} = o(x^{b}) \) for all nonnegative constants \(b > a\) 

 \(ln(x)=o(x^{\varepsilon}) \) for all \(\varepsilon > 0 \) 

 \(x^{c} = o(a^{x}) \) for any \(a, c \in \mathbb{R} \) with \(a > 1 \) 

 \(log(x)=o(x^{\epsilon}) \) for all \(\epsilon > 0 \) 

 

Let \(f(n) = O(g(n)) \). Why do we say \(\limsup\limits_{x \rightarrow \infty} \frac{f(x)}{g(x)} \) < \(\infty \) instead of \(lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} \) < \(\infty \)?

Exercise 3

&nbsp;It accommodates cases in which a limit of \(f(x)/g(x) \) does not exist.&nbsp;

&nbsp;The limit superior, \(limsup \), is a more strict limit compared to the ordinary limit, \(lim \).&nbsp;

&nbsp;\(O(g(x)) \) gives an upper bound instead of a lower bound on the growth of \(f(n) \).&nbsp;

&nbsp;\(f(x) \) can oscillate wildly as long as it never gets bigger than \(c \cdot g(x) \) for \(x > n \) and \(c > 0 \).&nbsp;