Let $0<\omega<\infty, \mu >0$ and $z \in \mathbb{R}.$ In my book, it is written that we have the following asymptotic behaviour:
i) Claim: $$\lim_{t \rightarrow \infty} \frac{z \omega}{\sqrt{t \mu}}=0.$$ ii) Claim: $$\lim_{t \rightarrow \infty} \left( \frac{z \omega}{\mu} \sqrt{\frac{t}{\mu}}+\frac{t}{\mu} \right)=\infty.$$
Can I proof it like that?
i) $$\lim_{t \rightarrow \infty} \frac{z \omega}{\sqrt{t \mu}}= \frac{z \omega}{\sqrt{ \mu}} \lim_{t \rightarrow \infty} \frac{1}{t^{1/2} }=0,$$ since $\lim_{t \rightarrow \infty} \frac{1}{t^n} =0$ for $n>0.$
ii) $$\lim_{t \rightarrow \infty} \left( \frac{z \omega}{\mu} \sqrt{\frac{t}{\mu}}+\frac{t}{\mu} \right)=\lim_{t \rightarrow \infty} \frac{z \omega}{\mu} \sqrt{\frac{t}{\mu}}+ \lim_{t \rightarrow \infty} \frac{t}{\mu} = \frac{z \omega}{\mu} \sqrt{\frac{1}{\mu}} \lim_{t \rightarrow \infty} \sqrt{t}+ \frac{1}{\mu} \lim_{t \rightarrow \infty} t=\infty.$$