Evaluation of $\lim_{x\to -\infty}(3/4)^x - (5/4)^x$

Question

I have to evaluate the following limit and I want to know if the below algebraic manipulation is legitimate.

$$\lim_{x\to -\infty}\left(\frac{3}{4}\right)^x - \left(\frac{5}{4}\right)^x$$ Can I change the limit to positive infinity, seeing as that would just be the reciprocal of the fractions as such?

$$\lim_{x\to \infty}\left(\frac{4}{3}\right)^x - \left(\frac{4}{5}\right)^x$$

Answer 1

Yes that’s a very nice way to proceed when dealing with limts to $-\infty$ in order to avoid confusion and now you can conclude that

$$\lim_{x\to \infty}\left(\frac{4}{3}\right)^x - \left(\frac{4}{5}\right)^x=(\infty-0)=\infty$$

since it is not an indeterminate form.

Answer 2

Your idea of doing $x\gets -x$ is very good, at least on psychological grounds.

You can also consider a more general situation, that hides the particular values involved, which hinder in the way: $$ \lim_{x\to\infty}(a^x-b^x) $$ with $a>b>0$. You can rewrite it as $$ \lim_{x\to\infty}a^x\left(1-\Bigl(\frac{b}{a}\Bigr)^x\right) $$ Since $0<b/a<1$, we have $$ \lim_{x\to\infty}\Bigl(\frac{b}{a}\Bigr)^x=0 $$ and therefore you have just to consider $\lim_{x\to\infty}a^x$. In your case $a=4/3>1$, so the limit is $\infty$.