VERY LARGE NUMBER

The quotient is near to $\left( \frac { 8 }{ 9 } \right) ^{ n }$ ,which might be near to zero if n is very large.

If we take:

$\frac{8^n+7^n+6^n}{9^n-8^n-7^n} = \frac{8^n(1 + (7/8)^n + (6/8)^n)}{8^n((9/8)^n - 1 - (7/8)^n)} = \frac{1 + (7/8)^n + (6/8)^n}{(9/8)^n - 1 - (7/8)^n}$

then the numerator $\rightarrow 1$ and the denominator $\rightarrow \infty$ as $n \rightarrow \infty$ . Hence $\lim_{n \rightarrow \infty} \frac{8^n+7^n+6^n}{9^n-8^n-7^n} \rightarrow \boxed{0}.$