Limit $\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}$ $$\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}=\lim_{x\rightarrow\infty}1-x+x\sqrt{\frac{2}{x^2}+\frac 2x+1}=\lim_{x\rightarrow\infty}1=1\neq2$$ as Wolfram Alpha state. Where I miss something?

For convention, let $\lim$ denote $\lim_{x\rightarrow \infty}$. I avoid big-O notation and power series here and perhaps it is clearer this way?
Now, $$\lim [1-x+\sqrt{2+2x+x^2}] \\
= \lim [1 + \frac{2+2x}{x + \sqrt{2+2x+x^2}}] \\
= \lim [1 + \frac{2+2x}{x(1+\sqrt{2/x^2+2/x+1})}] \\
= \lim [1 + \frac{2/x + 2}{1+\sqrt{2/x^2+2/x+1}}] \\
= 1 + \frac{2}{1+\sqrt{1}} = 2.$$
