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Question
\[\lim_{n \to \infty} \frac{n!}{\left( n + 1 \right)! + n!}\] is equal to
Options
\[\frac{1}{2}\]
0
2
1
Solution
0
\[\lim_{n \to \infty} \left[ \frac{n!}{\left( n + 1 \right)! + n!} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{n!}{\left( n + 1 \right) \times n! + n!} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{n!}{n!\left( n + 1 + 1 \right)} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{1}{n + 2} \right]\]
\[ = 0\]
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