$$\lim_{x \to \infty}[(8^n + 4^n)^\frac{1}{3} -2^n]
= \lim_{x \to \infty}\frac{(8^n+4^n-8^n)}{[(8^n+4^n)^{(\frac{1}{3})}]^2 +[(8^n+4^n)^{(\frac{1}{3})}][2^n]+[2^n]^2}$$
$$= \lim_{x \to \infty}\frac{4^n}{[(8^n+4^n)^{(\frac{1}{3})}]^2 +[(8^n+4^n)^{(\frac{1}{3})}][2^n]+[2^n]^2}$$
$$= \lim_{x \to \infty}\frac{1}{[(1+(\frac{1}{2})^n)^\frac{1}{3}]^2 +[(1+(\frac{1}{2})^n)^\frac{1}{3}][1]+1}$$
$$= \frac{1}{3}$$
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