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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?

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    $\begingroup$ Take $x_i = \cos(i)$ as a fine counter-example. $\endgroup$
    – Furrer
    Sep 22, 2016 at 20:46

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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 $-\infty$.

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