If $\lim_{i \to \infty}\frac{x_i}{y_i} = 0$ with $\lim_{i \to \infty} y_i = \infty$ what happens for $\lim_{i \to \infty} x_i$? If $\lim_{i \to \infty}\frac{x_i}{y_i} = 0$ with $\lim_{i \to \infty} y_i = \infty$ can we say that $\lim_{i \to \infty} x_i$ is either $\infty$, $-\infty$, or some constant $L$? Or can we not rule out other possibilities?

Here is another possibility.
Consider $x_{n} = (-1)^{n}n$ and $y_{n} = n^{2}$.
Then, $\lim_{n\rightarrow\infty}y_{n} = \infty$ and $\lim_{n\rightarrow\infty}\frac{x_{n}}{y_{n}} = 0$, but $x_{n}$ has subsequences going to $\infty$ and $-\infty$.
