<s>[INST] The limit of a function with sum of two roots. I need to find the limit of the following function:
$$\lim_{x\to-\infty} \left(\sqrt{x^2+2x}+\sqrt[3]{x^3+x^2}\right).$$
I derived it to the form of:
$$\lim_{x\to-\infty}\frac{(x^2+2x)^3-(x^3+x^2)^2}{\left(\sqrt{x^2+2x}-\sqrt[3]{x^3+x^2}\right)\left((x^2+2x)^2+(x^2+2x)\sqrt[3]{(x^3+x^2)^2}+\sqrt[3]{(x^3+x^2)^4}\right)},$$
hoping that I'd be able to simplify something but still have $\frac{0}{0}$ and don't see how to do that.
 [/INST] For $x<0$,
$$f(x)=-x(1+\frac{2}{x})^{\frac{1}{2}}+x(1+\frac{1}{x})^\frac{1}{3}$$
$$=-x-1+x+\frac{1}{3}+\frac{1}{x}\epsilon(x).$$
the limit is $$-\frac{2}{3}.$$
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