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