1. **State the problem:** Find the derivative of the function $f(x) = \sqrt{\frac{1}{x}}$ using the limit definition of the derivative. 2. **Recall the limit definition of the derivative:** $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Rewrite the function:** $$f(x) = \sqrt{\frac{1}{x}} = \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$$ 4. **Apply the limit definition:** $$f'(x) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}}{h}$$ 5. **Find a common denominator inside the numerator:** $$= \lim_{h \to 0} \frac{\frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x+h} \sqrt{x}}}{h} = \lim_{h \to 0} \frac{\sqrt{x} - \sqrt{x+h}}{h \sqrt{x+h} \sqrt{x}}$$ 6. **Rationalize the numerator:** Multiply numerator and denominator by $\sqrt{x} + \sqrt{x+h}$: $$= \lim_{h \to 0} \frac{(\sqrt{x} - \sqrt{x+h})(\sqrt{x} + \sqrt{x+h})}{h \sqrt{x+h} \sqrt{x} (\sqrt{x} + \sqrt{x+h})} = \lim_{h \to 0} \frac{x - (x+h)}{h \sqrt{x+h} \sqrt{x} (\sqrt{x} + \sqrt{x+h})}$$ 7. **Simplify the numerator:** $$= \lim_{h \to 0} \frac{-h}{h \sqrt{x+h} \sqrt{x} (\sqrt{x} + \sqrt{x+h})}$$ 8. **Cancel $h$ in numerator and denominator:** $$= \lim_{h \to 0} \frac{-1}{\sqrt{x+h} \sqrt{x} (\sqrt{x} + \sqrt{x+h})}$$ 9. **Evaluate the limit as $h \to 0$:** $$= \frac{-1}{\sqrt{x} \sqrt{x} (\sqrt{x} + \sqrt{x})} = \frac{-1}{x (2 \sqrt{x})} = -\frac{1}{2 x^{3/2}}$$ **Final answer:** $$f'(x) = -\frac{1}{2 x^{3/2}}$$