$\underset{n\to \mathrm{\infty }}{lim} \frac{a^n+b^n}{a^n-b^n}$ where $1<b<a$ is equal to

We have $1<b<a.$

$\therefore 0<\frac{b}{a}<1\Rightarrow \underset{n\to \mathrm{\infty }}{lim} {\left(\frac{b}{a}\right)}^n=0$

So, $\underset{n\to \mathrm{\infty }}{lim} \frac{a^n+b^n}{a^n-b^n}=\underset{n\to \mathrm{\infty }}{lim} \frac{1+{\left(\frac{b}{a}\right)}^n}{1-{\left(\frac{b}{a}\right)}^n}=\frac{1+0}{1-0}=1$