<s>[INST] The limit : $ \lim _{x \to \infty } \sqrt{x^2 +x} - \sqrt{x^2 +1} $ The limit: 
$ \lim _{x \to \infty } \sqrt{x^2 +x} - \sqrt{x^2 +1} $
(A) is 0 (B) is $\frac 1 2 $ (C) is 2 (D) does not exist
Is doing it with Binomial expansion and cancelling the terms only way? 
 [/INST] \begin{array} \\
\displaystyle \lim_{x \rightarrow \infty} \sqrt{x^2 + x} - \sqrt{x^2 + 1} &= \displaystyle \lim_{x \rightarrow \infty} \left(\sqrt{x^2 + x} - \sqrt{x^2 + 1}\right) \frac{\sqrt{x^2 + x} + \sqrt{x^2 + 1}}{\sqrt{x^2 + x} + \sqrt{x^2 + 1}} \\
&= \displaystyle \lim_{x \rightarrow \infty} \frac{(x^2 + x) - (x^2 + 1)}{\sqrt{x^2 + x} + \sqrt{x^2 + 1}} \\
&= \displaystyle \lim_{x \rightarrow \infty} \frac{x-1}{\sqrt{x^2 + x} + \sqrt{x^2 + 1}} \cdot \frac{(1/x)}{(1/x)} \\
&= \displaystyle \lim_{x \rightarrow \infty} \frac{1-1/x}{\sqrt{1 + 1/x} + \sqrt{1 + 1/x^2}}  \\
&= \displaystyle \frac{1}{2}
\end{array}
</s>