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;