1. ## Limits

Suppose $\lim_{x\to\infty} f(x)/g(x) = \lim_{x\to\infty} h(x)k(x)/g(x)$. Can you conclude from this that $\lim_{x\to\infty} k(x) = \lim_{x\to\infty} f(x)/h(x)$?

I make this step in a somewhat longer proof in my homework, but I am not sure if this is true in general. If it is not true in general, what conditions do I need on the functions?

2. To me, everything seems consistent with the Properties of Limits, assuming you meant to have parenthesis around all of your functions.
ie. $\lim_{x \to \infty}\left(f(x)/g(x) \right)$

Indeterminate Forms to understand what happens to these limits when dealing with dividing by 0 and infinity and such.

3. Thanks. The limits are indeed over everything that is to the right of it. I was indeed wondering if everything was correct with 0 and infinity.

4. consider the following functions:

$f(x) = \frac{1}{x^2}, g(x) = \frac{1}{x}, h(x) = f(x), k(x) = g(x)$. then

$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{h(x)k(x)}{g(x)} = 0$, however:

$\lim_{x \to \infty} k(x) = \lim_{x \to \infty} \frac{1}{x} = 0$, whereas $\lim_{x \to \infty} \frac{f(x)}{h(x)} = 1$.

one cannot apply the multiplicative/quotient properties of limits in this case, unless one is certain that the limits in the denominators are non-zero real numbers. in the example i have given, that is precisely the case for g(x) and h(x).

for another counter-example, suppose that:

$f(x) = x^2, g(x) = x, h(x) = f(x), k(x) = g(x)$.

again, we have:

$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{h(x)k(x)}{g(x)} = \infty$, but

$\lim_{x \to \infty} k(x) = \lim_{x \to \infty} x = \infty$, whereas $\lim_{x \to \infty} \frac{f(x)}{h(x)} = 1$.

suppose that: $\lim_{x \to \infty} f(x) = L, \lim_{x \to \infty} g(x) = M, \lim_{x \to \infty} h(x) = N, \lim_{x \to \infty} k(x) = P$.

to argue that L/M = NP/M implies L = NP, M cannot be 0, and must be finite.

and then to argue further that L = NP implies L/P = N, means P cannot be 0, or infinite. you can't divide by 0, or "cancel infinities".