Find the limit of this function Find
$$\lim_{n\to\infty}\left(1+x^n+\left(\frac {x^2}2\right)^n\right)^{\frac{1}{n}}$$
$(x\ge0)$.
Actually I could not find it when $x\ge1$, but when $0\leq x\lt1$ I found that the limit is $1$. Help me please.

$$\begin{align} \lim_{n\to\infty} \left(1+x^n+2^{-n}x^{2n}\right)^{1/n} \end{align}$$
Let's consider the roots of the quadratic: $x^n=\frac{-1\pm\sqrt{1-2^{2-n}}}{2^{1-n}}$
Thus, we have
$$\begin{align} &\lim_{n\to\infty} \left(1+x^n+2^{-n}x^{2n}\right)^{1/n} \\
=&\lim_{n\to\infty} (2^{-n})^{1/n}\left(x^n+\frac{1+\sqrt{1-2^{2-n}}}{2^{1-n}}\right)^{1/n}\left(x^n+\frac{1-\sqrt{1-2^{2-n}}}{2^{1-n}}\right)^{1/n} \\
=&\lim_{n\to\infty} \frac{1}{2}\left(x^n+\frac{2}{2^{1-n}}\right)^{1/n}\left(x^n+\frac{0}{2^{1-n}}\right)^{1/n} \\
=&\lim_{n\to\infty} \frac{x}{2}\left(x^n+2^n\right)^{1/n} \\
=&\frac{x}{2}\max(x,2)  \end{align}$$
