Evaluate this limit $\lim_{x\to - \infty}\frac{4^{x+3}-3^{x+2}-2^{x+1}}{4^{x+1}+3^{x+2}+2^{x+3}}$ Evaluate the limit
$$\lim_{x\to - \infty}\frac{4^{x+3}-3^{x+2}-2^{x+1}}{4^{x+1}+3^{x+2}+2^{x+3}}.$$
I've tried and it always comes out $\frac{0}{0}$, and l'Hopital doesn't seem to help me much here, what would you do?

Notice that the expressions $4^{x+3}$ and $3^{x+2}$ converge to zero more rapidly when $x\to-\infty$ than the expression $2^{x+1}$. Doing the same thing in the denominator we get
$$\lim_{x\to - \infty}\frac{4^{x+3}-3^{x+2}-2^{x+1}}{4^{x+1}+3^{x+2}+2^{x+3}}=\lim_{x\to - \infty}\frac{-2^{x+1}}{2^{x+3}}=-\frac14$$
