Chapter 1 - Limits and Continuity - 1.2 Computing Limits - Exercises Set 1.2 - Page 70: 38

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Given $$\lim_{x\to 0}\frac{\sqrt{x^2+4}-2}{x}$$ Then \begin{align*} \lim_{x\to 0}\frac{\sqrt{x^2+4}-2}{x}&=\lim_{x\to 0}\frac{\sqrt{x^2+4}-2}{x}\frac{\sqrt{x^2+4}+2}{\sqrt{x^2+4}+2}\\ &=\lim_{x\to 0}\frac{x^2+4-2}{x\sqrt{x^2+4}+2} \\ &=\lim_{x\to 0}\frac{x}{ \sqrt{x^2+4}+2}\\ &=0 \end{align*}