1. **State the problem:** Find the limit $$\lim_{h \to 0} \frac{5e^x - 5e^{x+h}}{3h}$$ without using a calculator. 2. **Rewrite the expression:** We can factor out 5 from the numerator: $$\lim_{h \to 0} \frac{5(e^x - e^{x+h})}{3h} = \frac{5}{3} \lim_{h \to 0} \frac{e^x - e^{x+h}}{h}$$ 3. **Rewrite the difference inside the limit:** Note that $$e^{x+h} = e^x e^h$$, so: $$\frac{e^x - e^{x+h}}{h} = \frac{e^x - e^x e^h}{h} = e^x \frac{1 - e^h}{h}$$ 4. **Use the known limit:** Recall that $$\lim_{h \to 0} \frac{e^h - 1}{h} = 1$$, so: $$\lim_{h \to 0} \frac{1 - e^h}{h} = \lim_{h \to 0} \frac{-(e^h - 1)}{h} = -1$$ 5. **Combine all parts:** $$\lim_{h \to 0} \frac{5(e^x - e^{x+h})}{3h} = \frac{5}{3} e^x (-1) = -\frac{5}{3} e^x$$ 6. **Final answer:** The limit equals $$-\frac{5}{3} e^x$$, which corresponds to option C.