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Question
The value of \[\lim_{x \to \infty} \frac{n!}{\left( n + 1 \right)! - n!}\]
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Solution
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\[\lim_{n \to \infty} \frac{n!}{\left( n + 1 \right)! - n!}\]
\[\text{ Dividing } N^r \text{ and } D^r \text{ by } n!: \]
\[ \lim_{n \to \infty} \frac{1}{\frac{\left( n + 1 \right)!}{n!} - 1}\]
\[ = \lim_{n \to \infty} \frac{1}{\frac{\left( n + 1 \right)n!}{n!} - 1}\]
\[ = \lim_{n \to \infty} \frac{1}{n + 1 - 1}\]
\[ = \lim_{n \to \infty} = \frac{1}{n} = 0\]
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