Chapter 1 - Limits and Continuity - 1.2 Computing Limits - Exercises Set 1.2 - Page 70: 37

$$\frac{1}{4}$$

Given $$\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x}$$ Then \begin{align*} \lim_{x\to 0}\frac{\sqrt{x+4}-2}{x}&=\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x}\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}\\ &=\lim_{x\to 0}\frac{x+4-2}{x\sqrt{x+4}+2} \\ &=\lim_{x\to 0}\frac{1}{ \sqrt{x+4}+2}\\ &=\frac{1}{4} \end{align*}