<s>[INST] Find $\lim_{x\to ∞} (\sqrt[3]{x^{3}+3x^{2}}-\sqrt{x^{2}-2x})$ without L'Hopital or Taylor series. 
$$\large \lim_{x\to ∞} (\sqrt[3]{x^{3}+3x^{2}}-\sqrt{x^{2}-2x})$$

My try is as follows:
$$\large \lim_{x\to ∞} (\sqrt[3]{x^{3}+3x^{2}}-\sqrt{x^{2}-2x})=$$$$ \lim_{x\to 
∞}x\left(\sqrt[3]{1+\frac{3}{x}}-\sqrt{1\ -\frac{2}{x}}\right)$$$$=\lim_{x\to ∞}x\lim_{x\to 
∞}\left(\sqrt[3]{1+\frac{3}{x}}-\sqrt{1\ -\frac{2}{x}}\right)$$
which is $∞×0$ , but clearly this zero is not exactly zero. I was thinking about generalized binomial 
theorem, but seems it will make the limit difficult, so how this kind of limits can be solved without using Taylor series or L'Hopital's rule?
 [/INST] Using high-school math:$$\lim_{x\to ∞} (\sqrt[3]{x^{3}+3x^{2}}-\sqrt{x^{2}-2x})=\lim_{x\to ∞} \frac{\sqrt[3]{(x^{3}+3x^{2})^2}-x^{2}+2x}{\sqrt[3]{x^{3}+3x^{2}}+\sqrt{x^{2}-2x}}=$$ $$ \lim_{x\to ∞} \frac{(x^{3}+3x^{2})^2-(x^{2}-2x)^3}{(\sqrt[3]{x^{3}+3x^{2}}+\sqrt{x^{2}-2x})(\sqrt[3]{(x^{3}+3x^{2})^4}+\sqrt[3]{(x^{3}+3x^{2})^2}(x^{2}-2x)+(x^2-2x)^2)}=$$ $$\lim_{x\to ∞} \frac{12x^5-3x^4+8x^3}{(\sqrt[3]{x^{3}+3x^{2}}+\sqrt{x^{2}-2x})(\sqrt[3]{(x^{3}+3x^{2})^4}+\sqrt[3]{(x^{3}+3x^{2})^2}(x^{2}-2x)+(x^2-2x)^2)}=$$ $$[\text{leaving the highest power}]=\lim_{x\to ∞} \frac{12x^5}{(x+x)(x^4+x^4+x^4)}=2$$
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