<s>[INST] Simplify $\frac {2\sqrt{x^2+x+1}-2-x} {x^2} $ In order to find the limit of the function at $0$ I need to simplify the fraction $$\frac {2\sqrt{x^2+x+1}-(2+x)} {x^2} $$
so that at $0$ it didn't look like $\frac 0 0$. I'm almost certain that it can be solved with a clever substitution of the variable but I didn't manage to find one.
I've also tried multiplying numerator and denominator by $2\sqrt{x^2+x+1}+2+x$, but without any success.
 [/INST] Hint. By Taylor expansion, it is easy to see that for $t\to 0$, 
$$\sqrt{1+t}=1+\frac{t}{2}-\frac{t^2}{8}+o(t^2).$$
Hence
$$2\sqrt{x^2+x+1}=2\sqrt{1+(x+x^2)}=2+x+x^2-\frac{(x+x^2)^2}{4}+o((x+x^2)^2)\\=2+x+\frac{3}{4}x^2+o(x^2).
$$
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