Marsh Barker
01/21/2024 · Middle School
\( \lim _ { n \rightarrow \infty } \frac { 7 } { n ! } \)
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Tutor-Verified Answer
Step-by-step Solution
To evaluate the limit
\[
\lim_{n \rightarrow \infty} \frac{7}{n!},
\]
we start by recognizing the behavior of \( n! \) (n factorial) as \( n \) approaches infinity. The factorial function grows very rapidly; specifically, \( n! \) increases much faster than any polynomial or exponential function.
As \( n \) becomes very large, \( n! \) tends to infinity. Therefore, we can analyze the limit as follows:
\[
\frac{7}{n!} \rightarrow 0 \quad \text{as} \quad n \rightarrow \infty.
\]
This is because the numerator (7) is a constant, while the denominator \( n! \) grows without bound. Thus, the fraction approaches zero.
Therefore, we conclude that:
\[
\lim_{n \rightarrow \infty} \frac{7}{n!} = 0.
\]
Thus, the final answer is
\[
\boxed{0}.
\]
Quick Answer
The limit is 0.
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