# Limit of 2^n/n! as n Approaches Infinity

• hanmoyuan
In summary, the limit of 2^n/n! as n approaches infinity is equal to 0. This can be calculated using the ratio test, and the expression 2^n/n! represents the sequence of terms in the infinite series for the function f(x) = 2^x. The reason for this limit being 0 is because the denominator grows much faster than the numerator, and it has significance in understanding the behavior of exponential functions for large values of x.
hanmoyuan
Find the limit of $$\frac{2^n}{n!}$$ when n approaches infinity.

hanmoyuan said:
Find the limit of $$\frac{2^n}{n!}$$ when n approaches infinity.
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## 1. What is the limit of 2^n/n! as n approaches infinity?

The limit of 2^n/n! as n approaches infinity is equal to 0. This means that as n gets bigger and bigger, the value of 2^n/n! will get closer and closer to 0.

## 2. How do you calculate the limit of 2^n/n! as n approaches infinity?

To calculate this limit, we can use the ratio test. This involves taking the limit of the ratio of the n+1 term to the nth term. In this case, the ratio would be (2^(n+1)/(n+1)!)/(2^n/n!), which simplifies to 2/(n+1). As n approaches infinity, this ratio will approach 0, indicating that the limit is also 0.

## 3. What does the expression 2^n/n! represent?

The expression 2^n/n! represents the sequence of terms that make up the infinite series for the function f(x) = 2^x. As n approaches infinity, the terms in this sequence get smaller and smaller, eventually approaching 0.

## 4. Why does the limit of 2^n/n! equal 0?

This limit equals 0 because the denominator (n!) grows much faster than the numerator (2^n) as n approaches infinity. This means that the value of the fraction will become smaller and smaller, approaching 0 as n gets larger.

## 5. What is the significance of the limit of 2^n/n! as n approaches infinity?

The significance of this limit is that it helps us understand the behavior of exponential functions as the input (in this case, n) gets larger and larger. It also allows us to approximate the value of these functions for very large values of x.

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