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Suppose $\{x_n\}$ and $\{y_n\}$ are sequences in $\mathbb{R}$ such that $$\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n = \infty$$ and $$\lim_{n \to \infty} \frac{x_n}{y_n} = q.$$ What can we say about $$\lim_{n \to \infty} x_n-y_nq?$$

What conditions can we put on $\{x_n\}$ and $\{y_n\}$ to get the limit of $0$?

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    $\begingroup$ Surely this is a typo. You must mean $x_n/y_n\to q$? That is, the denominator should be $y_n$, not $q_n$, right? $\endgroup$ – MPW Oct 14 '14 at 1:12
  • $\begingroup$ @MPW: Fixed. Thanks. $\endgroup$ – Sandeep Silwal Oct 14 '14 at 1:15
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Anything can happen - you might get lucky and have the second limit be 0, but not always. For example, consider $x_n=n^2+n$ and $y=n^2$. We have that $\lim\limits_{n\to\infty} \frac{x_n}{y_n}=1$ but $\lim\limits_{n\to\infty}x_n-y_n=\infty$.

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  • $\begingroup$ What extra conditions can we put on the sequences to have the limit be 0? $\endgroup$ – Sandeep Silwal Oct 14 '14 at 1:20

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