Puzzled by $\lim_{x \to +\infty} (\sqrt{x^2 + x} - x) = \lim_{x \to +\infty} \frac{x}{\sqrt{x^2 + x} + x}$

Question

In chapter 3, example 42 of Mathematical Analysis, vol 1, Zorich states that $$ \lim_{x \to +\infty} (\sqrt{x^2 + x} - x) = \lim_{x \to +\infty} \frac{x}{\sqrt{x^2 + x} + x}. $$ This equivalence is not at all clear to me. Is it an error? I would love to see a derivation or justification.

Answer 1

Just rationalise the numerator of the limit by multiplying both numerator and denominator by the conjugate of numerator. $$\begin{align}\lim_{x \to \infty} (\sqrt{x^2 + x} - x)\cdot \frac{\sqrt{x^2 + x} + x}{\sqrt{x^2 + x} + x} &= \lim_{x \to \infty} \frac{(\sqrt{x^2 + x} - x)(\sqrt{x^2 + x} + x)}{\sqrt{x^2 + x} + x}\\& =\lim_{x \to \infty} \frac{x^2 + x - x^2}{\sqrt{x^2 + x} + x}\\& =\lim_{x \to \infty} \frac{x}{\sqrt{x^2 + x} + x} \end{align}$$