SUMMARY
The limit of the sequence defined by the expression \(\lim_{n \rightarrow \infty} \frac{2^n}{n!} = 0\) is established using the definition of limits. By demonstrating that \(\frac{2^n}{n!} < \frac{4}{n}\) for sufficiently large \(n\), it is shown that this expression can be made less than any arbitrary \(\varepsilon > 0\) by choosing \(N = [\frac{4}{\varepsilon}] + 1\). Thus, the limit converges to zero as \(n\) approaches infinity.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with factorial notation and properties
- Knowledge of epsilon-delta definitions of limits
- Basic algebraic manipulation skills
NEXT STEPS
- Study the epsilon-delta definition of limits in more detail
- Learn about the growth rates of exponential functions versus factorial functions
- Explore convergence tests for sequences and series
- Investigate other limit examples involving factorials and exponentials
USEFUL FOR
Students of calculus, mathematicians, and anyone interested in understanding limits involving exponential and factorial functions.