If we have a limit of the form:
$$\lim_{x \to \infty} \frac{a(x)-b}{c(x)}$$
Is it always possible to write it as $$-\lim_{x \to \infty} \frac{b-a(x)}{c(x)}$$
Can this always be done, or must parts of the expression be/not be functions of $x$?
Edit: the question is motivated by the math in this answer.
This is an allowed property of limits. Recall
$$\lim_{x \to a} k\cdot f(x) = k\cdot \lim_{x \to a} f(x),$$
So, in your original limit, you can factor out the numerator of the fraction to be: $$a(x)-b=-1\cdot(b-a(x))$$
Now, using Eqn.1, you can simply take out the constant multiple of -1 from the original limit.
If limits $\lim_{x \to \infty} f(x)$ and $\lim_{x \to \infty} g(x)$ exist, then limit $\lim_{x \to \infty} f(x)g(x)$ also exists.
Take $f(x)=\frac{a(x)-b}{c(x)}$ and $g(x)=-1$.
So, by multiplying the initial limit twice with $-1$, you get the result you want.
