Which condition is stronger between (a) and (b)? In other words does (a) imply (b), or viceversa? Let $\{\lambda_n\}_n$ be a sequence of real positive numbers.
(a) $\lim_{n\rightarrow \infty}\frac{\lambda_n}{\sqrt{n}}=\lambda_0$
[that is equivalent to writing $\frac{\lambda_n}{\sqrt{n}}=O(1)$]
(b) $\lim_{n\rightarrow \infty}\frac{\lambda_n}{n}=0$
[that is equivalent to writing $\frac{\lambda_n}{n}=o(1)$]
(a) is not equivalent to $\lambda_n/\sqrt n = O(1)$, but it is sufficient for $\lambda_n/\sqrt{n} = O(1)$. It is possible that $\lambda_n/\sqrt{n} = O(1)$ but that the limit does not exist. Consider $\lambda_n = \sqrt n\sin n$, for example.
Now, if $\lambda_n/\sqrt n = O(1)$ then $\lambda_n/n = O(1/\sqrt n) = o(1)$. However if $\lambda_n = n^{3/4}$ then $\lambda_n/\sqrt n \to \infty$ but $\lambda_n/n \to 0$.
Thus $\lambda_n/\sqrt{n} = O(1) \implies \lambda_n / n = o(1)$, but the reverse is not true.