Limit of a sequence

Find the limit of the following sequence. \[ {\left({\left( \root{n}\of{0.5}+\frac{\root{n}\of{0.5}} {n} \right)}^{n}\right)}_{ n=1}^{\infty } \] Hint: The limit of the sequence \({\left({\left(1 + \frac{1} {n}\right)}^{n}\right)}_{n=1}^{\infty }\) is the Euler number \(\mathrm{e}\).

Find the limit. \[ \lim\limits_{n\to\infty}⁡\left(10^{-1} + 10^{-2} + \dots + 10^{-n} \right) \]

Find the following limit. \[ \lim\limits_{n\to\infty}⁡ \left( \frac{1+2+3+\dots+n}{4n^2-3}\right) \]

Find the following limit. \[ \lim _{n\to \infty } \frac{3 + 6 + 9 +\cdots +3n} {6 + 12 + 18 +\cdots +6n} \]