Limit of 10^n/n as n-> infinity

• ayandas
In summary, the limit of 10^n/n as n approaches infinity approaches infinity. This means that the value of the limit will continue to increase without bound as n gets larger and larger. To prove this, we can use the definition of a limit and manipulate the expression to show it can be made arbitrarily large. The limit cannot approach a finite number because the expression will continue to increase without bound. Replacing 10 with a different number will still result in the limit approaching infinity. This concept is useful in various fields of scientific research.
ayandas
Hi,

Can anyone please suggest a solution to the problem:

lim 10n/(n!)
n->(infinite)

When n is large, the numerator is much larger than the demoninator, so the limit is 0, surely?

Are you sure? When n is small, the numerator is larger, but when n gets large, the denominator is actually larger.

EDIT: Next time, put a question like this in the 'Homework Questions' section. That's where it should be, and you'll also probably get a response faster that way.

My bad, got the names the wrong way round ;)

No worries, just didn't want to confuse the OP!

1. What does the limit of 10^n/n approach as n approaches infinity?

The limit of 10^n/n as n approaches infinity approaches infinity. This means that the value of the limit will continue to increase without bound as n gets larger and larger.

2. How do you prove that the limit of 10^n/n as n approaches infinity is infinity?

To prove that the limit of 10^n/n as n approaches infinity is infinity, we can use the definition of a limit. We need to show that for any arbitrarily large number M, there exists a corresponding value of n where 10^n/n is greater than M. This can be done by manipulating the expression 10^n/n to show that it can be made arbitrarily large as n increases.

3. Can the limit of 10^n/n as n approaches infinity approach a finite number?

No, the limit of 10^n/n as n approaches infinity cannot approach a finite number. This is because the expression 10^n/n will continue to increase without bound as n gets larger, meaning it will never approach a fixed value.

4. What happens if we replace 10 with a different number in the expression 10^n/n as n approaches infinity?

If we replace 10 with a different number in the expression 10^n/n as n approaches infinity, the limit will still approach infinity. This is because any number raised to a large enough power will eventually become larger than n, causing the expression to increase without bound.

5. Is the limit of 10^n/n as n approaches infinity a useful concept in scientific research?

Yes, the concept of limits is widely used in scientific research, including in fields such as calculus, physics, and economics. The limit of 10^n/n as n approaches infinity is particularly useful in understanding the behavior of exponential growth and can be applied to various real-world scenarios.

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