I am trying to determine how to find the tightest upper and lower bounds when using the squeeze theorem. If there is a general technique to determining them I would appreciate the insight. Specifically I have this problem:
$$\lim_{n\to \infty} \sqrt[n]{2\left(\frac12\right)^n+\left(\frac23\right)^n+3\left(\frac12\right)^n}$$
I know that:
$$\sqrt[n]{\left(\frac23\right)^n}\le \sqrt[n]{2\left(\frac12\right)^n+\left(\frac23\right)^n+3\left(\frac12\right)^n} \le \sqrt[n]{6\left(\frac12\right)^n}$$
What steps were taken to find the upper bound?