Doubts regarding limits and logarithms

Lets say you are given this limit

$$\underset{n\to \mathrm{\infty}}{lim}(\mathrm{log}(n+{n}^{n}+{n}^{1/n})$$

That expression is equal to

$$\mathrm{log}(\underset{n\to \mathrm{\infty}}{lim}[n+{n}^{n}+{n}^{1/n}])$$

isn't it?

My question is if I could descompose the limit like this without changing the limit like this

$$\mathrm{log}(\underset{n\to \mathrm{\infty}}{lim}n+\underset{n\to \mathrm{\infty}}{lim}{n}^{n}+\underset{n\to \mathrm{\infty}}{lim}{n}^{1/n})$$

Could I?

Lets say you are given this limit

$$\underset{n\to \mathrm{\infty}}{lim}(\mathrm{log}(n+{n}^{n}+{n}^{1/n})$$

That expression is equal to

$$\mathrm{log}(\underset{n\to \mathrm{\infty}}{lim}[n+{n}^{n}+{n}^{1/n}])$$

isn't it?

My question is if I could descompose the limit like this without changing the limit like this

$$\mathrm{log}(\underset{n\to \mathrm{\infty}}{lim}n+\underset{n\to \mathrm{\infty}}{lim}{n}^{n}+\underset{n\to \mathrm{\infty}}{lim}{n}^{1/n})$$

Could I?