<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] $$ \lim \limits_{x \to \infty } \sqrt{x^2 +x} - \sqrt{x^2 +1}=\lim \limits_{x \to \infty }\dfrac{(\sqrt{x^2 +x} - \sqrt{x^2 +1})(\sqrt{x^2 +x} + \sqrt{x^2 +1})}{\sqrt{x^2+x}+\sqrt{x^2+1}}=\lim \limits_{x \to \infty } \dfrac{x-1}{\sqrt{x^2+x}+\sqrt{x^2+1}}=\lim \limits_{x \to \infty } \dfrac{1-\dfrac{1}{x}}{\sqrt{1+\dfrac{1}{x}}+\sqrt{1+\dfrac{1}{x^2}}}=\dfrac{1}{2} $$
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