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.

What you made is $\frac{a^{n+1}{+}b^{n+1}}{a^n+b^n}=a+\frac{b-a}{1+(a/b)^n}$ and using $\left( \frac{a}{b}\right)^n\to 0$, when $n\to\infty$, you obtain answer $b,$ so everything is correct.
Another possible way is
$$\frac{a^{n+1}{+}b^{n+1}}{a^n+b^n} = b\frac{1+\left( \frac{a}{b}\right)^{n+1}}{1+\left( \frac{a}{b}\right)^{n}}\to b,\text{ when }n\to\infty.$$
