1. **State the problem:** Calculate the limit $$\lim_{n \to \infty} \frac{3 + (-3)^n}{4^n}$$ by definition. 2. **Recall the limit definition and properties:** For sequences, if the numerator grows slower than the denominator or oscillates boundedly while the denominator grows without bound, the limit tends to zero. 3. **Analyze the numerator:** The term $3$ is constant. The term $(-3)^n$ oscillates between positive and negative values but its absolute value is $3^n$. 4. **Analyze the denominator:** The denominator is $4^n$, which grows exponentially faster than $3^n$. 5. **Rewrite the expression:** $$\frac{3 + (-3)^n}{4^n} = \frac{3}{4^n} + \frac{(-3)^n}{4^n} = \frac{3}{4^n} + \left(\frac{-3}{4}\right)^n$$ 6. **Evaluate each term's limit:** - Since $4^n$ grows without bound, $\lim_{n \to \infty} \frac{3}{4^n} = 0$. - Since $\left|\frac{-3}{4}\right| = \frac{3}{4} < 1$, $\lim_{n \to \infty} \left(\frac{-3}{4}\right)^n = 0$. 7. **Combine results:** $$\lim_{n \to \infty} \frac{3 + (-3)^n}{4^n} = 0 + 0 = 0$$ **Final answer:** $$\boxed{0}$$