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Question
Write the value of \[\lim_{n \to \infty} \frac{n! + \left( n + 1 \right)!}{\left( n + 1 \right)! + \left( n + 2 \right)!} .\]
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Solution
\[\lim_{n \to \infty} \left[ \frac{n! + \left( n + 1 \right)!}{\left( n + 1 \right)! + \left( n + 2 \right)!} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{n! + \left( n + 1 \right)n!}{\left( n + 1 \right)! + \left( n + 2 \right) \left( n + 1 \right)!} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{n! \left( 1 + n + 1 \right)}{\left( n + 1 \right)! \left( 1 + n + 2 \right)} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{n! \left( n + 2 \right)}{\left( n + 1 \right) n! \left( n + 3 \right)} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{n + 2}{\left( n + 1 \right) \left( n + 3 \right)} \right]\]
\[\text{ Degree of the denominator is greater than that of the numerator }.\]
\[ \Rightarrow 0\]
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