$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$

Question

I have a question:

$$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$$

Thanks for your help>

Answer 1

Since $|-2|<3$ then $(-2)^n=_\infty o(3^n)$ and then

$$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=\lim_{n \to \infty} \frac{3^n}{3^{n+1}}=\frac13$$

Answer 2

Hint: As $n$ goes to $\infty$, what is the ratio of $\large\frac{(-2)^n}{3^n}$. What does this ratio tell you about the importance of the addition?