Question:

**Step 1:** The limit to be evaluated is $\lim_{n \to \infty} \sqrt[n]{a^n + b^n}$. **Step 2:** Without loss of generality, assume $a \geq b > 0$. Factor out $a^n$ from the expression: $$ \sqrt[n]{a^n + b^n} = \sqrt[n]{a^n \left(1 + \left(\frac{b}{a}\right)^n\right)} = a \sqrt[n]{1 + \left(\frac{b}{a}\right)^n} $$ **Step 3:** Since $a \geq b > 0$, it follows that $0 < \frac{b}{a} \leq 1$. As $n \to \infty$, the term $\left(\frac{b}{a}\right)^n$ approaches $0$. Therefore, $1 + \left(\frac{b}{a}\right)^n \to 1 + 0 = 1$. **Step 4:** Let $x_n = 1 + \left(\frac{b}{a}\right)^n$. We have $\lim_{n \to \infty} x_n = 1$. For any sequence $x_n$ such that $\lim_{n \to \infty} x_n = L$ where $L > 0$, it is a standard property that $\lim_{n \to \infty} x_n^{1/n} = 1$. Applying this property to our expression: $$ \lim_{n \to \infty} \sqrt[n]{1 + \left(\frac{b}{a}\right)^n} = \lim_{n \to \infty} \left(1 + \left(\frac{b}{a}\right)^n\right)^{1/n} = 1 $$ Substituting this back into the limit from Step 2: $$ \lim_{n \to \infty} \sqrt[n]{a^n + b^n} = a \cdot 1 = a $$ If we had assumed $b > a$, a similar argument would yield $b$. Therefore, the value of the limit is $\max(a,b)$.