1. The problem asks to identify the function $f$ and the number $a$ for which the limit $$\lim_{h \to 0} \frac{\sqrt[5]{32 + h} - \sqrt[5]{32}}{h}$$ is the definition of the derivative $f'(a)$. 2. Recall the definition of the derivative: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ 3. Comparing, we see that: - $f(x) = \sqrt[5]{x} = x^{\frac{1}{5}}$ - $a = 32$ 4. To evaluate the limit, we find $f'(a)$ using the power rule: $$f'(x) = \frac{1}{5} x^{\frac{1}{5} - 1} = \frac{1}{5} x^{-\frac{4}{5}}$$ 5. Substitute $x = 32$: $$f'(32) = \frac{1}{5} \times 32^{-\frac{4}{5}}$$ 6. Simplify $32^{-\frac{4}{5}}$: Since $32 = 2^5$, $$32^{-\frac{4}{5}} = (2^5)^{-\frac{4}{5}} = 2^{5 \times -\frac{4}{5}} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16}$$ 7. Therefore, $$f'(32) = \frac{1}{5} \times \frac{1}{16} = \frac{1}{80}$$ 8. So the value of the limit is $\frac{1}{80}$. **Final answer:** $$\lim_{h \to 0} \frac{\sqrt[5]{32 + h} - \sqrt[5]{32}}{h} = \frac{1}{80}$$