deletions | additions       diff --git a/ratio.tex b/ratio.tex  index 5a05d94..3e91c85 100644  --- a/ratio.tex  +++ b/ratio.tex 
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\lim_{n\to\infty}\left|\:\frac{(n\:+\:2)\cdot x}{(n\:+\:4)}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:3)}{(n\:+\:1)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(n\:+\:2)\cdot x}{(n\:+\:4)}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:3)}{(n\:+\:1)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(n\:+\:3)(n\:+\:2)\cdot x}{(n\:+\:1)(n\:+\:4)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(n^2\:+\:2n\:+\:3n\:+\:6)\cdot x}{(n^2\:+\:4n\:+\:n\:+\:4)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(n^2\:+\:5n\:+\:6)\cdot x}{(n^2\:+\:5n\:+\:4)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{n^2\left(\frac{n^2}{n^2}\:+\:\frac{5n}{n^2}\:+\:\frac{6}{n^2}\right)\cdot x}{n^2\left(\frac{n^2}{n^2}\:+\:\frac{5n}{n^2}\:+\:\frac{4}{n^2}\right)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]  \end{align}  \end{document}