How do you determine the limit of #n^n/(n!)# as n approaches infinity?

Answer 1

#oo#

#n^n/(n!)=prod_{k=1}^n (n/k) = prod_{k=1}^{n-1}(n/k)#
but here each #n/k > 1#
then #lim_{n->oo}prod _{k=1}^{n-1}(n/k) = oo#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To determine the limit of n^n/(n!) as n approaches infinity, we can use the concept of the ratio test. By applying the ratio test, we can simplify the expression and evaluate the limit.

First, let's rewrite the expression as (n/n) * (n/n-1) * (n/n-2) * ... * (n/2) * (n/1).

Next, we can simplify each term by canceling out common factors. This results in n * (1/n-1) * (1/n-2) * ... * (1/2) * (1/1).

Now, we can observe that as n approaches infinity, each term in the expression (1/n-1), (1/n-2), ..., (1/2), (1/1) approaches zero.

Therefore, the limit of n^n/(n!) as n approaches infinity is zero.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7