1. **State the problem:** We need to find the limit of the function $$f(x) = 7 - 2e^{-3x}$$ as $$x$$ approaches infinity. 2. **Recall the limit rule for exponential decay:** When $$x \to \infty$$, the term $$e^{-3x}$$ approaches 0 because the exponent $$-3x$$ becomes very large negative, making $$e^{-3x}$$ very small. 3. **Apply the limit:** $$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left(7 - 2e^{-3x}\right) = 7 - 2 \cdot \lim_{x \to \infty} e^{-3x}$$ 4. **Evaluate the exponential limit:** $$\lim_{x \to \infty} e^{-3x} = 0$$ 5. **Substitute back:** $$7 - 2 \cdot 0 = 7$$ 6. **Final answer:** $$\boxed{7}$$ The limit of $$f(x)$$ as $$x$$ approaches infinity is 7.