evaluate $\lim _{x\to \infty} (3x^2-x^3)^{\frac{1}{3}}+x$ 
$$\lim _{x\to \infty} (3x^2-x^3)^{\frac{1}{3}}+x$$

can I look at $\lim\limits_{x\to \infty} (3^{\frac{1}{3}}x^{\frac{2}{3}}-x+x)$?

HINT:
$$\left(1+z\right)^{1/3}=1+\frac13 z+O(z^2) \tag 1$$
SPOILER ALERT:  Scroll over the highlighted area to reveal the full solution

 $$\begin{align}x+\left(3x^2-x^3\right)^{1/3}&=x-x\left(1-\frac{3x^2}{x^3}\right)^{1/3}\\\\&=x-x\left(1-\frac3x\right)^{1/3}\\\\&=x-x\left(1-\frac1x+O\left(\frac1{x^2}\right)\right)\\\\&=1+O\left(\frac1x\right)\to 1\,\,\text{as}\,\,x\to \infty\end{align}$$

