Limits of an indeterminate form $\lim_{t\to\infty} (a+b(-m)^t)/(c+d(-m)^{t-1})$

Question

I'm trying to solve the limit of the following indeterminate form:

$$\lim_{t\to\infty} \frac{a+b(-m)^t}{c+d(-m)^{t-1}}$$

where $t=1, 2, 3, \cdots$ denotes time and all the coefficients are positive rational numbers.

In particular, I would like to consider two cases: i) $0<m<1$ and ii) $1<m$.

Any comments would be greatly appreciated!

Answer 1

Hint :

i) $0\lt m\lt 1$

Since $-1\lt -m\lt 0$, we have $(-m)^{t}\to 0$ as $t\to \infty$.

ii) $1\lt m$

Since $-1\lt\frac{1}{-m}\lt 0$, we have $\frac{1}{(-m)^{t-1}}\to 0$ as $t\to\infty$. Now, divide both the numerator and the denominator by $(-m)^{t-1}$.

Answer 2

if $m>0$ we have $-m<0$ and if $x \in \mathbb{R}$ what kind of number is $$(-m)^{x}$$? in this form makes the limit no sense.