1. **Problem:** Find the limit of the sequence $\left\{(n^2 + 2)^{1/n}\right\}_{n=1}^\infty$. 2. **Formula and rules:** For sequences of the form $a_n^{1/n}$, the limit can often be found by considering the dominant term inside the root and using properties of limits and logarithms. 3. **Work:** \begin{align*} \lim_{n \to \infty} (n^2 + 2)^{1/n} &= \lim_{n \to \infty} e^{\frac{1}{n} \ln(n^2 + 2)} \\ &= e^{\lim_{n \to \infty} \frac{\ln(n^2 + 2)}{n}}. \end{align*} 4. Since $\ln(n^2 + 2) \sim \ln(n^2) = 2 \ln n$ as $n \to \infty$, we have $$\lim_{n \to \infty} \frac{\ln(n^2 + 2)}{n} = \lim_{n \to \infty} \frac{2 \ln n}{n} = 0$$ because $\ln n$ grows slower than $n$. 5. Therefore, $$\lim_{n \to \infty} (n^2 + 2)^{1/n} = e^0 = 1.$$