1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{3n}$$. 2. **Rewrite the expression:** Note that $$\frac{n+1}{n} = 1 + \frac{1}{n}$$, so the limit becomes $$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3n}$$. 3. **Recall the important limit:** We know that $$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$$, where $e$ is Euler's number approximately equal to 2.71828. 4. **Use the property of exponents:** Since the exponent is $3n$, we can write: $$\left(1 + \frac{1}{n}\right)^{3n} = \left[\left(1 + \frac{1}{n}\right)^n\right]^3$$. 5. **Apply the limit:** Taking the limit as $n \to \infty$, $$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3n} = \left(\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\right)^3 = e^3$$. 6. **Final answer:** $$\boxed{e^3}$$. This means the expression grows and approaches $e^3$ as $n$ becomes very large.