Evaluating a Limit by Rationalizing

Consider the following statements regarding the evaluation of the limit $\lim_{x\to0} \frac{\sqrt{x+4}-2}{x}$.

I. The expression can be simplified by rationalizing the numerator: $\frac{\sqrt{x+4}-2}{x} \cdot \frac{\sqrt{x+4}+2}{\sqrt{x+4}+2} = \frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{\sqrt{x+4}+2}$ for $x \neq 0$.

II. Applying the limit to the simplified expression yields $\lim_{x\to0} \frac{1}{\sqrt{x+4}+2} = \frac{1}{\sqrt{0+4}+2} = \frac{1}{4}$.

III. The limit does not exist because the function defined by $f(x)=\frac{\sqrt{x+4}-2}{x}$ is not defined at $x=0$.

Which of the above statements is or are true?