Find the limit of each of the following sequences; justify your answers.

(i)

$$\frac{1+2+\ldots+n}{n^{2}}$$

(ii)

$$\sqrt[n]{n}$$

(iii)

$$\left(a^{n}+b^{n}\right)^{1 / n} \quad \text { with } \quad 0<a \leqslant b$$