<s>[INST] Can anyone tell me if my proof is right? Let $0<a<b,$ compute $\lim_{n\to \infty}{\frac{a^{n+1}{+}b^{n+1}}{a^n+b^n}}$
My proof are followings :
Assuming that $r=b-a>0,$ so $b=a+r.$ So the original formula equals to $\frac{a\cdot a^{n}{+}{(a+r)\cdot}b^{n}}{a^n+b^n}=a+\frac{r\cdot b^n}{a^n+b^n}=a+\frac{r}{1+(a/b)^n},$ so it can limit to $a+r=b.$
From my point of view, I think that the results should contain $a,$ however my result contains no $a.$ I am a self learner, I don't know wheather my result is right. Hope that someone can help me out.
 [/INST] Your proof is correct. Here's another way to see why the limit is $b$:
Notice that
\begin{align*}
\frac{a^{n+1}+b^{n+1}}{a^n+b^n} &= a^n\cdot\frac{a}{a^n+b^n}+b^n\cdot\frac{b}{a^n+b^n}\\
&= \frac{1}{\frac{1}{a^n}}\cdot\frac{a}{a^n+b^n}+\frac{1}{\frac{1}{b^n}}\cdot\frac{b}{a^n+b^n}\\
&= \frac{a}{1+\left(\frac{b}{a}\right)^n}+ \frac{b}{1+\left(\frac{a}{b}\right)^n}
\end{align*}
Since $b>a$, it's easy to see that $\left(\frac{b}{a}\right)^n\to\infty$ and $\left(\frac{a}{b}\right)^n\to 0$ as $n\to\infty$, so
$$\lim\limits_{n\to\infty}\left(\frac{a}{1+\left(\frac{b}{a}\right)^n}+ \frac{b}{1+\left(\frac{a}{b}\right)^n}\right)=0+\frac{b}{1}=b$$
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