Calculate $\lim_{x\to\infty}\frac{x^2}{x+1}-\sqrt{x^2+1}$ I am stuck on a limit of the indeterminate form $\infty-\infty$. I have tried many approaches, such as multiplying with conjugates etc. and I am unable to find a solution. I suspect that there is an elementary trick that I am plainly missing right here. Can anybody give me a hint or solution as to solve
$$\lim_{x\to\infty}\frac{x^2}{x+1}-\sqrt{x^2+1}$$

1) $x^2=(x+1)-1)^2=$
$(x+1)^2-2(x+1)+1$;
$\dfrac{x^2}{x+1}=(x+1)-2 +\dfrac{1}{x+1};$
2) $(x^2+1)^{1/2}=x(1+1/(x^2))^{1/2}=$
$ x +O(1/x)$;
3) $\dfrac{x^2}{x+1} -(x^2+1)^{1/2}= $
$ -1+O(1/x)$;
