Limit of root of x squared plus x minus x

Question

Find $\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+x}-x\right)$ .

$\tfrac12$ . Multiply by the conjugate: $\sqrt{x^2+x}-x=\dfrac{x}{\sqrt{x^2+x}+x}=\dfrac{1}{\sqrt{1+1/x}+1}\to\tfrac12$ .

Adapted from Crack, Heard on the Street, Q1.66, p.21; question and answer rewritten by SelfMadeQuant.com.

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