Answer
converges to $0$
Work Step by Step
As we know that when $x \gt 0$, $\lim\limits_{n \to \infty}x^n=0$
Let $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{(\dfrac{10}{11})^n}{(\dfrac{9}{10})^n+(\dfrac{11}{12})^n}$
This implies that $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{(\dfrac{12}{11} \cdot \dfrac{10}{11})^n}{(\dfrac{12}{11} \cdot \dfrac{9}{10})^{(n)}+(\dfrac{12}{11} \cdot \dfrac{11}{12})^{(n)}}=\dfrac{0}{(0+1)}$
so, $a_n=0$
Thus, $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} converges to $0$.
