<s>[INST] How to calculate $\lim \limits_{x \to 0}{\frac{\sqrt{1 + x + x^2} - 1}{x}}$? I try to calculate $\lim \limits_{x \to 0}{\frac{\sqrt{1 + x + x^2} - 1}{x}}$. I've got $\frac{\sqrt{1 + x + x^2} - 1}{x} = \sqrt{\frac{1}{x^2} + \frac{1}{x} + x} - \frac{1}{x}$ but I don't know what to do next.
 [/INST] $$\sqrt{1 + x + x^2} = 1 + \dfrac{1}{2}(x+x^2) + \cdots = 1 + \dfrac{1}{2}x + \cdots$$ so the 
$$\lim \limits_{x \to 0}{\frac{\sqrt{1 + x + x^2} - 1}{x}} = 
\lim \limits_{x \to 0}\dfrac{1 + 1/2 x + \cdots - 1}{x} = \dfrac{1}{2}$$
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