$\lim_{x\to\infty} \frac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$ How can I solve this limit.  (Here $x$ belongs to natural numbers $\Bbb{N}$.) 
$$
\lim_{x\to\infty} \dfrac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$$
My try:  I tried using L'Hospital, expansions of all terms using Taylor's series, and did work out.  I was just thinking if there is any simpler method?

Of course asymptotics yields the most elegant way
since $a^x \pm b^x \sim_\infty a^x$ if $a > b > 0$
we have that the limit is equivalent to
$$\lim_{x\to\infty}\frac{5\cdot 5^x}{27 \cdot 9^x} = \lim_{x\to\infty}\frac{5}{27}\cdot \left(\frac{5}{9}\right)^x = 0$$
