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Tomorrow I have an exam on mathematical analysis and I solving exams task from earlier years, and here is:

Find limit of: $$\lim_{n \to \infty} \frac{(4n^3 + 1)(4n - 2)!n\sin{\frac{2}{n}}}{(4n + 1)!+3} $$

I checked in Wolfram Alpha and Maxima that it is $\frac{1}{8}$ but how to get this? I have tried almost everything I remember but still cannot find solution.

Firstly, sorry for the title, but I have no idea how to describe this shorter.

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Step 1: convince yourself that the $3$ in the denominator can safely be ignored.

Step 2: deal with the $n\sin(2/n)$ as one piece, and with the rest of it as another.

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