The equation is Lim (2^n)/sqrt(n!) = 0, where Lim represents the limit as n approaches infinity, 2^n represents 2 raised to the power of n, and sqrt(n!) represents the square root of the factorial of n.
To prove this limit, we can use the Squeeze Theorem. First, we can show that (2^n)/sqrt(n!) <= (2^n)/(n^(n/2)). Then, we can show that (2^n)/(n^(n/2)) approaches 0 as n approaches infinity. Therefore, by the Squeeze Theorem, we can conclude that Lim (2^n)/sqrt(n!) = 0.
This limit is important because it helps us understand the behavior of the function (2^n)/sqrt(n!) as n approaches infinity. It also has applications in various fields such as computer science, physics, and engineering.
One example is in the analysis of algorithms. The time complexity of certain algorithms can be represented by the function (2^n)/sqrt(n!), and by proving its limit is 0, we can conclude that the algorithm has a polynomial time complexity, making it more efficient.
When proving this limit, we assume that n is approaching infinity and that n is a positive integer. Additionally, this limit may not hold true for complex numbers or negative integers. It is important to also consider any possible exceptions or edge cases when applying this limit in real-world scenarios.