Question

Q: If 0<a<b, determine lim((a^(n+1)+b^(n+1))/(a^n+b^n ))

Answer 1

Answer on Question #64533 – Math – Real Analysis

Question

If $0 < a < b$, determine $\lim_{n\to \infty}\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}$.

Solution

$$\begin{array}{l} \lim _ {n \to \infty} \frac {a ^ {n + 1} + b ^ {n + 1}}{a ^ {n} + b ^ {n}} = \lim _ {n \to \infty} \frac {\frac {a ^ {n + 1} + b ^ {n + 1}}{b ^ {n}}}{\frac {a ^ {n} + b ^ {n}}{b ^ {n}}} = \lim _ {n \to \infty} \frac {b + \frac {a ^ {n}}{b ^ {n}} a}{1 + \frac {a ^ {n}}{b ^ {n}}} = \frac {\lim _ {n \to \infty} \left(b + \frac {a ^ {n}}{b ^ {n}} a\right)}{\lim _ {n \to \infty} \left(1 + \frac {a ^ {n}}{b ^ {n}}\right)} = \frac {\lim _ {n \to \infty} b + \lim _ {n \to \infty} \frac {a ^ {n}}{b ^ {n}} a}{\lim _ {n \to \infty} 1 + \lim _ {n \to \infty} \frac {a ^ {n}}{b ^ {n}}} = \\ = \frac {b + a \lim _ {n \to \infty} \left(\frac {a}{b}\right) ^ {n}}{1 + \lim _ {n \to \infty} \left(\frac {a}{b}\right) ^ {n}} = \frac {b + a \cdot 0}{1 + 0} = b. \end{array}$$

If $0 < a < b$, then $\frac{a}{b} < 1$ and $\lim_{n\to \infty}\left(\frac{a}{b}\right)^n = 0$.

Answer: b.

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